C# – Sql server real datatype, what is the C# equivalent

string is an alias in C# for System.String.
So technically, there is no difference. It's like int vs. System.Int32.

As far as guidelines, it's generally recommended to use string any time you're referring to an object.

e.g.

string place = "world";

Likewise, I think it's generally recommended to use String if you need to refer specifically to the class.

e.g.

string greet = String.Format("Hello {0}!", place);

This is the style that Microsoft tends to use in their examples.

It appears that the guidance in this area may have changed, as StyleCop now enforces the use of the C# specific aliases.

Syntax:

ALTER TABLE {TABLENAME} 
ADD {COLUMNNAME} {TYPE} {NULL|NOT NULL} 
CONSTRAINT {CONSTRAINT_NAME} DEFAULT {DEFAULT_VALUE}
WITH VALUES

Example:

ALTER TABLE SomeTable
        ADD SomeCol Bit NULL --Or NOT NULL.
 CONSTRAINT D_SomeTable_SomeCol --When Omitted a Default-Constraint Name is autogenerated.
    DEFAULT (0)--Optional Default-Constraint.
WITH VALUES --Add if Column is Nullable and you want the Default Value for Existing Records.

Notes:

Optional Constraint Name:
If you leave out CONSTRAINT D_SomeTable_SomeCol then SQL Server will autogenerate
    a Default-Contraint with a funny Name like: DF__SomeTa__SomeC__4FB7FEF6

Optional With-Values Statement:
The WITH VALUES is only needed when your Column is Nullable
    and you want the Default Value used for Existing Records.
If your Column is NOT NULL, then it will automatically use the Default Value
    for all Existing Records, whether you specify WITH VALUES or not.

How Inserts work with a Default-Constraint:
If you insert a Record into SomeTable and do not Specify SomeCol's value, then it will Default to 0.
If you insert a Record and Specify SomeCol's value as NULL (and your column allows nulls),
    then the Default-Constraint will not be used and NULL will be inserted as the Value.

Notes were based on everyone's great feedback below.
Special Thanks to:
    @Yatrix, @WalterStabosz, @YahooSerious, and @StackMan for their Comments.