The proper structure for this scenario is a SubClass / Inheritance model, and is nearly identical to the concept I proposed in this answer: https://dba.stackexchange.com/questions/71741/heterogeneous-ordered-list-of-value/72319#72319.

The model proposed in this question is actually quite close in that the `Animal` entity contains the type (i.e. `race`) and the properties that are common across all types. However, there are two minor changes that are needed:

1. Remove the Cat_ID and Dog_ID fields from their respective entities:

 The key concept here is that _everything_ is an `Animal`, regardless of `race`: `Cat`, `Dog`, `Elephant`, and so on. Given that starting point, any particular `race` of `Animal` doesn't truly need a separate identifier since:

 1. the `Animal_ID` is unique
 1. the `Cat`, `Dog`, and any additional `race` entities added in the future do not, by themselves, fully represent any particular `Animal`; they only have meaning when used in combination with the information contained in the parent entity, `Animal`.

 Hence, the `Animal_ID` property in the `Cat`, `Dog`, etc entities is both the PK and the FK back to the `Animal` entity.

1. Differentiate between types of `breed`:

 Just because two properties share the same name does not _necessarily mean_ that those properties are the same, even if the name being the same _implies_ such a relationship. In this case, what you really have is actually `CatBreed` and `DogBreed` as seperate "types"

Initial Notes
---
1. The SQL is specific to Microsoft SQL Server (i.e. is T-SQL). Meaning, be careful about datatypes as they are not the same across all RDBMS's. For example, I am using `VARCHAR` but if you need to store anything outside of the standard ASCII set, you should really use `NVARCHAR`.
1. The ID fields of the "type" tables (`Race`, `CatBreed`, and `DogBreed`) are _not_ auto-incrementing (i.e. IDENTITY in terms of T-SQL) because they are application constants (i.e. they are part of the application) that are static lookup values in the database and are represented as `enum`s in C# (or other languages). If values are added, they are added in controlled situations. I reserve the use of auto-increment fields for user data that comes in via the application.
1. The naming convention I use is to name each subclass table starting with the main class name followed by the subclass name. This helps organize the tables as well as indicates clearly (without looking at the FKs) the relationship of the subclass table to the main entity table.
1. Please see "Final Edit" section at the end for a note regarding Views.


"Breed" as "Race"-Specific Approach
---
![Breed As Race-specific Diagram][1]  
This first set of tables are the lookup / types tables:

<!-- language: lang-sql -->

    CREATE TABLE Race
    (
      RaceID INT NOT NULL PRIMARY KEY
      RaceName VARCHAR(50) NOT NULL
    );
    
    CREATE TABLE CatBreed
    (
      CatBreedID INT NOT NULL PRIMARY KEY,
      BreedName VARCHAR(50),
      CatBreedAttribute1 INT,
      CatBreedAttribute2 VARCHAR(10)
      -- other "CatBreed"-specific properties as needed
    );
    
    CREATE TABLE DogBreed
    (
      DogBreedID INT NOT NULL PRIMARY KEY,
      BreedName VARCHAR(50),
      DogBreedAttribute1 TINYINT
      -- other "DogBreed"-specific properties as needed
    );


This second listing is the main "Animal" entity:

<!-- language: lang-sql -->

    CREATE TABLE Animal
    (
      AnimalID INT NOT NULL IDENTITY(1, 1) PRIMARY KEY,
      RaceID INT NOT NULL, -- FK to Race
      Name VARCHAR(50)
      -- other "Animal" properties that are shared across "Race" types
    );

    ALTER TABLE Animal
      ADD CONSTRAINT [FK_Animal_Race]
      FOREIGN KEY (RaceID)
      REFERENCES Race (RaceID);

This third set of tables are the complimentary sub-class entities that complete the definition of each `Race` of `Animal`:

<!-- language: lang-sql -->

    CREATE TABLE AnimalCat
    (
      AnimalID INT NOT NULL PRIMARY KEY, -- FK to Animal
      CatBreedID INT NOT NULL, -- FK to CatBreed
      HairColor VARCHAR(50) NOT NULL
      -- other "Cat"-specific properties as needed
    );
    
    ALTER TABLE AnimalCat
      ADD CONSTRAINT [FK_AnimalCat_CatBreed]
      FOREIGN KEY (CatBreedID)
      REFERENCES CatBreed (CatBreedID);

    ALTER TABLE AnimalCat
      ADD CONSTRAINT [FK_AnimalCat_Animal]
      FOREIGN KEY (AnimalID)
      REFERENCES Animal (AnimalID);
    
    
    CREATE TABLE AnimalDog
    (
      AnimalID INT NOT NULL PRIMARY KEY, -- FK to Animal
      DogBreedID INT NOT NULL, -- FK to DogBreed
      HairColor VARCHAR(50) NOT NULL
      -- other "Dog"-specific properties as needed
    );
    
    ALTER TABLE AnimalDog
      ADD CONSTRAINT [FK_AnimalDog_DogBreed]
      FOREIGN KEY (DogBreedID)
      REFERENCES DogBreed (DogBreedID);
    
    ALTER TABLE AnimalDog
      ADD CONSTRAINT [FK_AnimalDog_Animal]
      FOREIGN KEY (AnimalID)
      REFERENCES Animal (AnimalID);

_The model using a shared `breed` type is shown after the "Additional Notes" section._

Additional Notes
---
1. The concept of `breed` seems to be a focal point for confusion. It was suggested by jcolebrand (in a comment on the question) that `breed` is a property shared across the different `race`s, and the other two answers have it integrated as such in their models. This is a mistake, however, because the values for `breed` are not shared across the different values of `race`. Yes, I am aware that the two other proposed models attempt to solve this issue by making `race` a parent of `breed`. While that technically solves the relationship issue, it doesn't help solve the overall modeling question of what to do about non-common properties, nor how to handle a `race` that does not have a `breed`. But, in the case that such a property were guaranteed to exist across all `Animal`s, I will include an option for that as well (below).
1. The models proposed by vijayp and DavidN (which seem to be identical) do not work because:
  1. They either
     1. do not allow for non-common properties to be stored (at least not for individual instances of any `Animal`), or
     1. require that all properties for all `race`s be stored in the `Animal` entity which is a very flat (and nearly non-relational) way of representing this data. Yes, people do this all of the time, but it means having many NULL fields per row for the properties that are not meant for that particular `race` AND knowing which fields per row are associated with the particular `race` of that record.
   1. They do not allow for adding a `race` of `Animal` in the future that does not have `breed` as a property. And even if ALL `Animal`s have a `breed`, that wouldn't change the structure due to what has been previously noted about `breed`: that `breed` is dependent on the `race` (i.e. `breed` for `Cat` is not the same thing as `breed` for `Dog`).

"Breed" as Common- / Shared- Property Approach
---
![enter image description here][2]  
Please note:

1. The SQL below can be run in the same database as the model presented above:

  1. The `Race` table is the same
  1. The `Breed` table is new
  1. The three `Animal` tables have been appended with a `2`
1. Even with `Breed` being a now common property, it does not seem right not to have `Race` noted in the main/parent entity (even if it is technically relationally correct). So, both `RaceID` and `BreedID` are represented in `Animal2`. In order to prevent a mismatch between the `RaceID` noted in `Animal2` and a `BreedID` that is for a different `RaceID`, I have added a FK on both `RaceID, BreedID` that references a UNIQUE CONSTRAINT of those fields in the `Breed` table. I usually despise pointing a FK to a UNIQUE CONSTRAINT, but here is one of the few valid reasons to do so. A UNIQUE CONSTRAINT is logically an "Alternate Key", which makes it valid for this use. Please also note that the `Breed` table still has a PK on just `BreedID`.
   1. The reason for not going with just a PK on the combined fields and no UNIQUE CONSTRAINT is that it would allow for the same `BreedID` to be repeated across different values of `RaceID`.
   1. The reason for not switching which the PK and UNIQUE CONSTRAINT around is that this might not be the only usage of `BreedID`, so it should still be possible to reference a specific value of `Breed` without having the `RaceID` available.
1. While the following model does work, it has two potential flaws regarding the concept of a shared `Breed` (and are why I prefer the `Race`-specific `Breed` tables).
  1. There is an implicit assumption that ALL values of `Breed` have the same properties. There is no easy way in this model to have disparate properties between `Dog` "breeds" and `Elephant` "breeds". However, there still is a way to do this, which is noted in the "Final Edit" section.
  1. There is no way to share a `Breed` across more than one race. I am not sure if that is desirable to do (or maybe not in the concept of animals but possibly in other situations that would be using this type of model), but it is not possible here.

<!-- language: lang-sql -->

    CREATE TABLE Race
    (
      RaceID INT NOT NULL PRIMARY KEY,
      RaceName VARCHAR(50) NOT NULL
    );
    
    CREATE TABLE Breed
    (
      BreedID INT NOT NULL PRIMARY KEY,
      RaceID INT NOT NULL, -- FK to Race
      BreedName VARCHAR(50)
    );
    
    ALTER TABLE Breed
      ADD CONSTRAINT [UQ_Breed]
      UNIQUE (RaceID, BreedID);
    
    ALTER TABLE Breed
      ADD CONSTRAINT [FK_Breed_Race]
      FOREIGN KEY (RaceID)
      REFERENCES Race (RaceID);
    
    CREATE TABLE Animal2
    (
      AnimalID INT NOT NULL IDENTITY(1, 1) PRIMARY KEY,
      RaceID INT NOT NULL, -- FK to Race, FK to Breed
      BreedID INT NOT NULL, -- FK to Breed
      Name VARCHAR(50)
      -- other properties common to all "Animal" types
    );
    
    ALTER TABLE Animal2
      ADD CONSTRAINT [FK_Animal2_Race]
      FOREIGN KEY (RaceID)
      REFERENCES Race (RaceID);
    
    -- This FK points to the UNIQUE CONSTRAINT on Breed, _not_ to the PK!
    ALTER TABLE Animal2
      ADD CONSTRAINT [FK_Animal2_Breed]
      FOREIGN KEY (RaceID, BreedID)
      REFERENCES Breed (RaceID, BreedID);
    
    
    CREATE TABLE AnimalCat2
    (
      AnimalID INT NOT NULL PRIMARY KEY, -- FK to Animal
      HairColor VARCHAR(50) NOT NULL
    );
    
    ALTER TABLE AnimalCat2
      ADD CONSTRAINT [FK_AnimalCat2_Animal2]
      FOREIGN KEY (AnimalID)
      REFERENCES Animal2 (AnimalID);
    
    CREATE TABLE AnimalDog2
    (
      AnimalID INT NOT NULL PRIMARY KEY,
      HairColor VARCHAR(50) NOT NULL
    );
    
    ALTER TABLE AnimalDog2
      ADD CONSTRAINT [FK_AnimalDog2_Animal2]
      FOREIGN KEY (AnimalID)
      REFERENCES Animal2 (AnimalID);

<br>

Final Edit (hopefully ;-)
---
1. Regarding the possibility (and then difficulty) of handling disparate properties between types of `Breed`, it _is_ possible to employ the same subclass / inheritance concept but with `Breed` as the main entity. In this setup the `Breed` table would have the properties common to all types of `Breed` (just like the `Animal` table) and `RaceID` would represent the type of `Breed` (same as it does in the `Animal` table). Then you would have subclass tables such as `BreedCat`, `BreedDog`, and so on. For smaller projects this _might_ be considered "over-engineering", but it is being mentioned as an option for situations that would benefit from it.
1. For both approaches, it sometimes helps to create Views as short-cuts to the full entities. For example, consider:

        CREATE VIEW Cats AS
           SELECT  an.AnimalID,
                   an.RaceID,
                   an.Name,
                   -- other "Animal" properties that are shared across "Race" types
                   cat.CatBreedID,
                   cat.HairColor
                   -- other "Cat"-specific properties as needed
           FROM    Animal an
           INNER JOIN  AnimalCat cat
                   ON  cat.AnimalID = an.AnimalID
           -- maybe add in JOIN(s) and field(s) for "Race" and/or "Breed"
1. While not part of the logical entities, it is fairly common to have audit fields in the tables to at least get a sense of when the records are being inserted and updated. So in practical terms:
  1. A `CreatedDate` field would be added to the `Animal` table. This field is not needed in any of the subclass tables (e.g. `AnimalCat`) as the rows being inserted for both tables should be done at the same time within a transaction.
  1. A `LastModifiedDate` field would be added to the `Animal` table and all subclass tables. This field gets updated only if that particular table is updated: if an update occurs in `AnimalCat` but not in `Animal` for a particular `AnimalID`, then only the `LastModifiedDate` field in `AnimalCat` would be set.


  [1]: https://i.sstatic.net/GrBRU.jpg
  [2]: https://i.sstatic.net/wNFTB.jpg