Store query result in a variable using in PL/pgSQL

I think you're looking for SELECT select_expressions INTO:

select test_table.name into name from test_table where id = x;

That will pull the name from test_table where id is your function's argument and leave it in the name variable. Don't leave out the table name prefix on test_table.name or you'll get complaints about an ambiguous reference.

The usual pattern is EXISTS(subselect):

BEGIN
  IF EXISTS(SELECT name
              FROM test_table t
             WHERE t.id = x
               AND t.name = 'test')
  THEN
     ---
  ELSE
     ---
  END IF;

This pattern is used in PL/SQL, PL/pgSQL, SQL/PSM, ...

To assign a single variable, you can also use plain assignment in a PL/pgSQL code block, with a scalar subquery to the right:

name := (SELECT t.name from test_table t where t.id = x);

Effectively the same as SELECT INTO like @mu already provided, with subtle differences:

• SELECT INTO is slightly faster in my tests on Postgres 14.
  (Plain assignment of a constant, without involving SELECT, is 10x faster, still.)
• SELECT INTO also sets the special variable FOUND, while plain assignment does not. You may want one or the other.
• SELECT INTO can also assign multiple variables at once. See:
  • SELECT INTO with more than one attribution

Notably, this works, too:

name := t.name from test_table t where t.id = x;

A SELECT statement without leading SELECT. But I would not use this hybrid. Better use one of the first two, clearer, documented methods, as @Pavel commented.