How can I verify an object is an instance of a specific class?

I have a method which accepts a PDO object as an argument, to allow the user to use an existing connection rather then the method to open a new one, and save resources:

public static function databaseConnect($pdo = null) {

I am aware of is_object() to check if the argument is an object, but I want to check if $pdo is a PDO object, and not just an object.

Because the user can easily enter (by mistake?) a different kind of object, a mysqli or such, and the entire script will break apart.

In short: How can I check a variable for a specific type of object?

You can use instanceof:

if ($pdo instanceof PDO) {
    // it's PDO
}

Be aware though, you can't negate like !instanceof, so you'd instead do:

if (!($pdo instanceof PDO)) {
    // it's not PDO
}

Also, looking over your question, you can use object type-hinting, which helps enforce requirements, as well as simplify your check logic:

function connect(PDO $pdo = null)
{
    if (null !== $pdo) {
        // it's PDO since it can only be
        // NULL or a PDO object (or a sub-type of PDO)
    }
}

connect(new SomeClass()); // fatal error, if SomeClass doesn't extend PDO

Typed arguments can be required or optional:

// required, only PDO (and sub-types) are valid
function connect(PDO $pdo) { }

// optional, only PDO (and sub-types) and 
// NULL (can be omitted) are valid
function connect(PDO $pdo = null) { }

Untyped arguments allow for flexibility through explicit conditions:

// accepts any argument, checks for PDO in body
function connect($pdo)
{
    if ($pdo instanceof PDO) {
        // ...
    }
}

// accepts any argument, checks for non-PDO in body
function connect($pdo)
{
    if (!($pdo instanceof PDO)) {
        // ...
    }
}

// accepts any argument, checks for method existance
function connect($pdo)
{
    if (method_exists($pdo, 'query')) {
        // ...
    }
}

As for the latter (using method_exists), I'm a bit mixed in my opinion. People coming from Ruby would find it familiar to respond_to?, for better or for worse. I'd personally write an interface and perform a normal type-hint against that:

interface QueryableInterface
{ 
    function query();
}

class MyPDO extends PDO implements QueryableInterface { }

function connect(QueryableInterface $queryable) { }

However, that's not always feasible; in this example, PDO objects are not valid parameters as the base type doesn't implement QueryableInterface.

It's also worth mentioning that values have types, not variables, in PHP. This is important because null will fail an instanceof check.

$object = new Object();
$object = null;
if ($object instanceof Object) {
    // never run because $object is simply null
}

The value loses it's type when it becomes null, a lack of type.