I would like to select a row based on some criteria, but if no row
  matches the criteria, I would like to return the first row
Shorter (and correct)
You don't actually need a WHERE clause at all:
SELECT street, zip, city
FROM   address
ORDER  BY street !~~ 'Test%', ord
LIMIT  1;
!~~ is just the Postgres operator for NOT LIKE. You can use either. Note that by inverting the logic (NOT LIKE instead of LIKE), we can now use default ASCsort order and NULLs sort last, which may be important. Read on.
This is shorter (but not necessarily faster). It is also subtly different (more reliable) than the currently accepted answer by @Gordon.
When sorting by a boolean expression you must understand how it works:
The currently accepted answer uses ORDER BY <boolean expression> DESC, which would sort NULLs first. In such a case you should typically add NULLS LAST:
If street is defined NOT NULL this is obviously irrelevant, but that has not been defined in the question. (Always provide the table definition.) The currently accepted answer avoids the problem by excluding NULL values in the WHERE clause.
Some other RDBMS (MySQL, Oracle, ..) don't have a proper boolean type like Postgres, so we often see incorrect advice from people coming from those products.
Your current query (as well as the currently accepted answer) need the WHERE clause - or at least NULLS LAST. With the different expression in ORDER BY neither is necessary.
More importantly, yet, if multiple rows have a matching street (which is to be expected), the returned row would be arbitrary and could change between calls - generally an undesirable effect. This query picks the row with the smallest ord to break ties and produces a stable result.
This form is also more flexible in that it does not rely on the existence of a row with ord = 0. Instead, the row with the smallest ord is picked either way.
Faster with index
(And still correct.)
For big tables, the following index would radically improve performance of this query:
CREATE INDEX address_street_pattern_ops_idx ON address(street text_pattern_ops);
Detailed explanation:
Depending on undefined details it may pay to add more columns to the index.
The fastest query using this index:
(
SELECT street, zip, city
FROM   address
WHERE  street LIKE 'Test%'
ORDER  BY ord  -- or something else?
-- LIMIT 1  -- you *could* add LIMIT 1 in each leg
)
UNION ALL
(
SELECT street, zip, city
FROM   address
ORDER  BY ord
-- LIMIT 1  -- .. but that's not improving anything in *this* case
)
LIMIT  1
BTW, this is a single statement.
This is more verbose, but allows for a simpler query plan. The second SELECT of the UNION ALL is never executed if the first SELECT produces enough rows (in our case: 1). If you test with EXPLAIN ANALYZE, you'll see (never executed) in the query plan.
Details:
Evaluation of UNION ALL
In reply to Gordon's comment. Per documentation:
Multiple UNION operators in the same SELECT statement are evaluated
  left to right, unless otherwise indicated by parentheses.
Bold emphasis mine.
And LIMIT makes Postgres stop evaluating as soon as enough rows are found. That's why you see (never executed) in the output of EXPLAIN ANALYZE.
If you add an outer ORDER BY before the final LIMIT this optimization is not possible. Then all rows have to be collected to see which might sort first.