What's the deal with speaking in tongues?

So you may be familiar with this Christian practice. It may also seem extremely strange for a group of people to be making sounds something like "Walla walla durka durka durka" and thinking that it's ok. You wouldn't be alone in thinking that. Even Paul the Apostle said:

If, therefore, the whole church comes together and all speak in tongues, and outsiders or unbelievers enter, will they not say that you are out of your minds? -- 1 Corinthians 14:23

But even for all the admitted weirdness, it is something that has a Biblical basis. '1 Corinthians 12-14' covers much of what tongues is about, but it also crops up elsewhere too.
The first time is in 'Acts 2' when the apostles gather for Pentecost. In verse 4 the Holy Spirit comes to them and allows them to speak in tongues. The following section details how there were Jews around from all over the world:

Parthians and Medes and Elamites and residents of Mesopotamia, Judea and Cappadocia, Pontus and Asia, Phrygia and Pamphylia, Egypt and the parts of Libya belonging to Cyrene, and visitors from Rome, both Jews and proselytes, Cretans and Arabians—we hear them telling in our own tongues the mighty works of God. -- Acts 2:9-11

The people from these various nations could understand what the apostles were saying in their own languages.

And at this sound the multitude came together, and they were bewildered, because each one was hearing them speak in his own language. -- Acts 2:6

So, we see very clearly that in this instance, the gift of tongues was given so that the people around the apostles could clearly understand the Gospel message.
Then it goes on: