Select * 
  from tableA
 where case 
             when @Type = 'Lname' then Lname 
             when @Type = 'fName' then Fname 
             when @Type is null then .....
             else .....
          end = @Value
 

--====== First Create TableA and add indexes: ======
Create Table TableA (
	ID int identity(1,1) primary key
	, Lname varchar(255)
	, Fname varchar(255)
)
Create index idxTableA_Fname on TableA(Fname)
Create index idxTableA_Lname on TableA(Lname)
go

--====== Now Load it with data:  ======
Insert into TableA (Lname, Fname)
 Select [name], [name]
 from master.sys.syscolumns
go

--====== Create the IF based stored procedure ======
Create Proc spGet_TableA(@type as varchar(20), @value varchar(255))
 AS
Begin
	If @Type = 'Lname'
	Begin
	select * from tableA
	where Lname = @Value
	End
	else If @Type = 'Fname'
	Begin
	select * from tableA
	where Fname = @Value
	end
	else if @type is null
	begin select * from TableA
	end
End
go

--====== Create the CASE based stored procedure ======
Create Proc spGet_TableA_case(@type as varchar(20), @value varchar(255))
 AS
Begin
	Select *
	  from tableA
	 where case
		 when @Type = 'Lname' then Lname
		 when @Type = 'fName' then Fname
		 when @Type is null then Null
		end = @Value
End
go

--====== Now Execute Both ======
--(actually, you should select these lines and user "Control-L"
--  to display the query plans)
EXEC spGet_TableA 'Lname', 'id'
go
EXEC spGet_TableA_case 'Lname', 'id'
go

create table tableA (Lname varchar(10), Fname varchar(10)) 

insert into tableA (Lname,Fname) select 'a1','b1' union select 'a2','b2'

--select * from tableA

declare @Type varchar(10),
            @value varchar(10),
            @sqlstr nvarchar(100);

set @Type = 'Lname';
set @value = 'a2';

set @sqlstr = 'select * from tableA ';

If @Type = 'Lname'
Begin
	set @sqlstr = @sqlstr + ' where Lname = ''' + @value + ''''
End
else If @Type = 'Fname'
Begin
	set @sqlstr = @sqlstr + ' where Fname = ''' + @value + '''';
end

print @sqlstr
exec sp_executesql 
		@stmt	= @sqlstr,
		@params	= N'@value varchar(255)',
		@value	= @value;

SELECT  *
FROM    TableA
WHERE   @Type = 'Lname'
AND     Lname = @Value

UNION   ALL

SELECT  *
FROM    TableA
WHERE   @Type = 'Fname'
AND     Fname = @Value

UNION   ALL

SELECT  *
FROM    TableA
WHERE   @Type IS NULL;