Can people speak in tongues today? The Bible refers to people speaking in tongues--most notably at Pentecost. Is this something that legitimately happens today?

I've been in church services where speaking in tongues has happened, but it always seems weird and creepy, and those doing it seem like kooks. How can I know if this is legitimate, or if I'm right in thinking it's strange?
Random babbling isn't "speaking in tongues" as the Apostles did on the Day of Pentecost.  The Apostles preached, and everyone present heard the message in their native language.  This is the purpose of speaking in tongues: to facilitate communication.  In 1 Corinthians 14 Paul warns very strongly about speaking in tongues simply for its own sake, since it doesn't accomplish any useful purpose.

As for modern occurrences, I've heard plenty of stories of missionaries and other preachers in situations where they did not possess sufficient language skills in the language of the people being taught gaining the temporary ability to express themselves clearly when it was necessary.  I don't have any references to cite right at the moment, but yes, it does still happen today.