First, I'll note that what you'll hear is typical of a DX station making brief contacts. There's not a lot of chat back and forth, but the goal here is to make as many contacts as possible. Second, I picked this clip (recorded earlier today) because the station I was listening to happened to have an exceptionally strong signal and conditions were very good. (For those of you interested, the DX station was using a 2KW amplifier into a 6 element cubical quad. My station is an Icom 756 Pro II with a G5RV antenna up about 10 m in the backyard. I worked him a few minutes before this recording was made.)
With that said, here's the link to the audio file that I'll be describing: http://k2dbk.com/rl3a.mp3 (you may need to right-click and save that to your computer, or you may just be able to click on it to play, depending on how your computer is set up.)
I'll give a start time in seconds for each description to help you follow along, here's what you're hearing:
:00 - "Q R Zed stateside, Radio LimaThree Alpha" - QRZ (hams pronounce the letter "Z" like "Zed" because it's so similar to other letters like C, etc.) is a sort of shorthand for "Who is calling me?". He uses the term "stateside" because he's just interested in making contacts with stations in the United States (though often that really includes Canada and Mexico as well.) Radio Lima Three Alpha are the radio phonetics for the callsign RL3A, who is the DX station that I referred to earlier. I'll explain more about him later, but what's happening here is that he's saying "This is RL3A and I'm ready for another contact".
:03 - At this point various stations are giving their callsigns phonetically (kind of like kids raising their hands in class and saying "pick me, pick me!"). Because of the way radio propagation works, you can't hear everyone calling him, but you can hear Whiskey Alpha Eight Lima Oscar Whiskey (WA8LOW) along with what sounds like a bunch of other people calling all at once. (In fact, that's exactly what's happening).
:07 - "Victor Echo United, what's the prefix?" Although we heard WA8LOW, RL3A has heard part of a callsign that ends in VUE and he's asking for the beginning of it.
:10 - First a bit of just noise, then "Roger, roger, Victor Echo Three Victor Echo United fifty-nine, QSL?" The noise (most of which I've edited out) is where VE3VEU is giving his complete callsign to RL3A. The reason you can't hear VE3VEU is because of propagation. That's a station in Canada and his signal is probably passing right over me, but was mostly likely very strong as heard by RL3A. RL3A acknowledges that he's heard the complete callsign and gives him a standard signal report, 59. The report is given using the RST (Readability, Signal, Tone) system, but in many cases a simple 59 report is used where the exact value of the report isn't important. QSL is a shorthand way of saying "Did you get the information that I sent you?"
:16 - More noise while VE3VUE is talking, then at about :21 "Seventy Three Bill, good luck Q R Zed Radio Lima Three Alpha" Seventy-Three (73) is another ham radio "code" which means "best regards" and is a common way to say "so long" at the end of a contact. If you've been keeping track, you've figured out that the next part is RL3A asking "who wants to be next?"
:25 - More stations calling, then "Whiskey Delta Eight Japan Papa something fifty-nine, over". This is pretty much the same as the previous contact, but in this case RL3A sent the 59 signal report right away. He's got most of WD8JP's call but thinks he might be missing a letter. "Over", as you might expect, just means that he's telling the other station to go ahead and talk.
:34 - I cut quite a bit of the noise out here since it was rather long, and then we hear "QSL John, I am Dima, Delta Italy Mike Alpha and the QTH Moscow. Thank you John for the QSO 73 good luck". In this case, RL3A is using QSL to acknowledge that he has heard the information sent (it can be used either as a question, as in the clip started at :10, or as an answer). Obviously the WD8 station operator has said his name is John (and it turns out that the complete station call was in fact WD8JP, that's why RL3A didn't respond with the full callsign again, since he had received it correctly the first time), and RL3A's name is Dima, which he spells phonetically. You have probably guessed that QTH is a shorthand for "location", and Dima is located in Moscow. He then closes out the contact with the usual "so long" and after that (though not recorded), he repeated the "loop" of working stations.
I hope you've found this informative, if a bit lengthy. If for some reason you have a problem downloading the MP3 file (it's a bit over 500k bytes in size), please let me know and I'll help you out.