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The proper structure for this scenario is a SubClass / Inheritance model, and is nearly identical to the concept I proposed in this answer: Heterogeneous ordered list of value.

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
2. 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"

Below is the SQL to construct the basic model. Please note:

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.
2. 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 enums 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.
3. 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.

This model is using the race-specific breed tables concept. This first set of tables are the lookup / types tables:

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:

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:

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);

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 races, 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 Animals, I will include an option for that as well.
2. The models proposed by vijayp and DavidN (which seem to be identical) do not work because:
3. They either
   1. do not allow for non-common properties to be stored (at least not for individual instances of any Animal), or
   2. require that all properties for all races 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.
4. 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 Animals 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).