When a person speaks in tongues, do they understand what they are saying?

In order to understand a person who has the gift of speaking tongues, you would either need a person with the gift of interpretation of tongues, or the person speaking might have the gift. The person whom is speaking tongues really isn't speaking, more so they are letting the Holy Spirit within them guild their words into the perfect prayer in a language unknown to them.

If you don't understand what is coming out of your mouth, then it is not the gift of tongues.

"The gift of tongues by the power of the Holy Ghost in the Church is for the benefit of the servants of God to preach to unbelievers, as on the day of Pentecost." (Teachings, p. 195)

By using the gift of tongues, on the day of Pentacost, the apostles were able to speak in their own tongue and be understood by people of many tongues. (Acts 2:1-18)

As far as speaking in tongues, Paul addresses the subject throughout 1 Corinthians, and we also see it in Acts and in Mark (all quotes from the ESV)

1 Cor 14:2-4

For one who speaks in a tongue speaks not to men but to God; for no one understands him, but he utters mysteries in the Spirit. On the other hand, the one who prophesies speaks to people for their upbuilding and encouragement and consolation. The one who speaks in a tongue builds up himself, but the one who prophesies builds up the church.

1 Cor 14:27-28

If any speak in a tongue, let there be only two or at most three, and each in turn, and let someone interpret. But if there is no one to interpret, let each of them keep silent in church and speak to himself and to God.

Acts 2:1-13, to me, is one of the best examples:

When the day of Pentecost arrived, they were all together in one place. 2 And suddenly there came from heaven a sound like a mighty rushing wind, and it filled the entire house where they were sitting. 3 And divided tongues as of fire appeared to them and rested on each one of them. 4 And they were all filled with the Holy Spirit and began to speak in other tongues as the Spirit gave them utterance.

5 Now there were dwelling in Jerusalem Jews, devout men from every nation under heaven. 6 And at this sound the multitude came together, and they were bewildered, because each one was hearing them speak in his own language. 7 And they were amazed and astonished, saying, “Are not all these who are speaking Galileans? 8 And how is it that we hear, each of us in his own native language? 9 Parthians and Medes and Elamites and residents of Mesopotamia, Judea and Cappadocia, Pontus and Asia, 10 Phrygia and Pamphylia, Egypt and the parts of Libya belonging to Cyrene, and visitors from Rome, 11 both Jews and proselytes, Cretans and Arabians—we hear them telling in our own tongues the mighty works of God.” 12 And all were amazed and perplexed, saying to one another, “What does this mean?” 13 But others mocking said, “They are filled with new wine.”

We see that tongues, as showed by the Pentacost, are all real world languages and can be understood. However, in most cases there is to be an interpreter (or there is one) in order to understand what is being said. That doesn't mean that the person speaking can't have the gift of interpretation, however one can logically conclude you couldn't have both the gift of tongues and interpretation as to avoid counterfeit gifts. This, as well as Paul going over how "the parts" of the Body are appointed and his description of the gifts, we see there is never reference or alluding to anyone possessing more than one of these gifts at a time.

1 Corinthians 12:4-11

4 Now there are varieties of gifts, but the same Spirit; 5 and there are varieties of service, but the same Lord; 6 and there are varieties of activities, but it is the same God who empowers them all in everyone. 7 To each is given the manifestation of the Spirit for the common good. 8 For to one is given through the Spirit the utterance of wisdom, and to another the utterance of knowledge according to the same Spirit, 9 to another faith by the same Spirit, to another gifts of healing by the one Spirit, 10 to another the working of miracles, to another prophecy, to another the ability to distinguish between spirits, to another various kinds of tongues, to another the interpretation of tongues. 11 All these are empowered by one and the same Spirit, who apportions to each one individually as he wills.

I would find it reasonable and logical to, without a person gifted with interpretation, say that a person that knows what they said were not displaying a legitimate gift. We see, at least in Paul's example, that speaking in tongues is speaking to God and should be interpreted. Also, keep in mind and as noted, that the Pentacost shows us these languages are very much real. Some will use 1 Cor. 13:1 to say there is a "special, angelic language". While this may be true, we are not told so. Analyzing this Scripture with the verses and even chapters surrounding shows us this is one of Paul's metaphors about how important LOVE is in the Church:

"If I speak with the tongues of men and of angels, but do not have love, I have become a noisy gong or a clanging cymbal."

Obviously Paul, nor anyone to our knowledge, became a gong or a cymbal.

The most relevant passage I can find in the Bible is this:

1 Corinthians 14:2 (ESV)
For one who speaks in a tongue speaks not to men but to God; for no one understands him, but he utters mysteries in the Spirit.

I'd venture a guess that if a person is "uttering mysteries in the Spirit", they don't understand what they're saying.