What is the CURRENT_DATE or current date value function for SQL Server?

You can use either GETDATE (return type datetime) or SYSDATETIME (return type datetime2), with the difference being the precision up to nanoseconds for SYSDATETIME().

Example:

SELECT GETDATE() fn_GetDate, SYSDATETIME() fn_SysDateTime

Results:

fn_GetDate                 fn_SysDateTime
2018-06-27 10:31:18.963    2018-06-27 10:31:18.9659170

See Date and Time Data Types and Functions (Transact-SQL) in the product documentation.

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For completeness, SQL Server also recognises CURRENT_DATE as mentioned in the question, as an ODBC scalar function:

SELECT {fn CURRENT_DATE()};

This returns varchar(10), so would need an explicit cast or convert to the date data type:

SELECT CONVERT(date, {fn CURRENT_DATE()});

The built-in functions are recommended over ODBC scalar functions.

CAST (... to date) with GETDATE() or SYSDATETIME()

The best way is

SELECT CAST( GETDATE() AS date );

By extension in SQL Server, you can cast SYSDATETIME() to date,

SELECT CAST( SYSDATETIME() AS date );

The docs on SYSDATETIME (Transact-SQL) show some more examples,

SELECT CONVERT (date, SYSDATETIME())  
    ,CONVERT (date, SYSDATETIMEOFFSET())  
    ,CONVERT (date, SYSUTCDATETIME())  
    ,CONVERT (date, CURRENT_TIMESTAMP)  
    ,CONVERT (date, GETDATE())  
    ,CONVERT (date, GETUTCDATE());  

/* All returned 2007-04-30 */  

There may be advantages CAST (GETDATE() AS date), as GETDATE() natively returns less precision.