How can I count the numbers of rows that a MySQL query returned?

In the event you have to solve the problem with simple SQL you might use an inline view.

select count(*) from (select * from foo) as x;

Getting total rows in a query result...

You could just iterate the result and count them. You don't say what language or client library you are using, but the API does provide a mysql_num_rows function which can tell you the number of rows in a result.

This is exposed in PHP, for example, as the mysqli_num_rows function. As you've edited the question to mention you're using PHP, here's a simple example using mysqli functions:

$link = mysqli_connect("localhost", "user", "password", "database");

$result = mysqli_query($link, "SELECT * FROM table1");
$num_rows = mysqli_num_rows($result);

echo "$num_rows Rows\n";

Getting a count of rows matching some criteria...

Just use COUNT(*) - see Counting Rows in the MySQL manual. For example:

SELECT COUNT(*) FROM foo WHERE bar= 'value';

Get total rows when LIMIT is used...

If you'd used a LIMIT clause but want to know how many rows you'd get without it, use SQL_CALC_FOUND_ROWS in your query, followed by SELECT FOUND_ROWS();

SELECT SQL_CALC_FOUND_ROWS * FROM foo
   WHERE bar="value" 
   LIMIT 10;

SELECT FOUND_ROWS();

For very large tables, this isn't going to be particularly efficient, and you're better off running a simpler query to obtain a count and caching it before running your queries to get pages of data.

SELECT SQL_CALC_FOUND_ROWS *
FROM   table1
WHERE  ...;

SELECT FOUND_ROWS();

FOUND_ROWS() must be called immediately after the query.

If your SQL query has a LIMIT clause and you want to know how many results total are in that data set you can use SQL_CALC_FOUND_ROWS followed by SELECT FOUND_ROWS(); This returns the number of rows A LOT more efficiently than using COUNT(*)
Example (straight from MySQL docs):

mysql> SELECT SQL_CALC_FOUND_ROWS * FROM tbl_name
    -> WHERE id > 100 LIMIT 10;
mysql> SELECT FOUND_ROWS();