Sure, I'll link to Wikipedia articles, but I'll provide the cliffnotes version...
Textus Receptus (TR) - It's a Latin phrase meaning "received text." It's a collection of Greek manuscripts (roughly 6) that was used in translating Luther's Bible, Tyndale's translation, and eventually the King James version of the Bible when it comes to the New Testament. It was incomplete, with some pieces of the text being back-translated from the Latin Vulgate when it was first compiled by Erasmus. It is of the Byzantine family (think Eastern Roman Empire origin) and falls later in date (IE: more recent). For the time when it was used, it was basically the best they had. We've since found many more copies and many more much older copies. The oldest date on the TR texts (to my knowledge) is 11th or 12th century. So they're old, but still at or over 1,000 years removed from the original autographs.
For KJV-onlyists, this is the manuscript group, nothing else will do. The argument basically centers around God preserving his Word and that the Textus Receptus is the most pure preserved Word.
http://en.wikipedia.org/wiki/Textus_Receptus
Majority Text (MT) - The majority text is Byzantine in nature as well. In a nutshell, the New Testament has probably the most manuscripts available of any ancient work. The majority of these are Byzantine and date from the 5th century through the 16th-17th century, having been preserved by the church. Don't let it fool you, though, as these texts do not all agree 100%. There are variations within the MT type. The earliest church Father to quote the Byzantine version was John Chrysostom who lived at the end of the 4th and beginning of the 5th centuries. The oldest Byzantine reading text is the Syriac Peshitta, which dates from the 4th century, but it contains readings also from another text type which I will hit in the next category. Basically, though, you can see that there is a missing 300-400 year gap in between the New Testament being written and the earliest Byzantine texts.
The MT doesn't have as much support - though you'll see some in Reformed and maybe Baptist circles argue for it. The NKJV, while a revision of the KJV, leaned on the MT for various readings that it believed the KJV essentially got wrong. There is an Orthodox Bible and some other minor versions that use or consult the MT, but it does not have widespread use. The only major versions outside the NKJV would be the WEB and I *think* YLT (Young's Literal) uses the MT.
The HCSB would have probably been based on the MT had the original Editor, Farstad, survived.
http://en.wikipedia.org/wiki/Majority_Text
Critical Text (CT) - The CT is the modern scholarly consensus text. For a brief while, we started out with the Alexandrian test type, consisting of about 30 manuscripts, which gives us the oldest manuscripts. Among these are the Codex Vaticanus and Codex Sinaiticus, both of which are old. This family is dated to the 3rd century and moves forward from there. They're called Alexandrian because they generally come from Egypt...it probably has something to do with the dry climate which is much more favorable. Church fathers from the 2nd century beyond tend to quote this version.
It's now called the Critical Text because textual critics have tried to use the vast manuscript array to whittle it down to the closest text to the originals. All modern versions are based off of this text, such as the NIV, ESV, HCSB, NASB, etc. This would include many Reformed, Mainline, Evangelical, Baptist, etc.
http://en.wikipedia.org/wiki/Alexandrian_text-type
http://en.wikipedia.org/wiki/Critical_text
The rub comes in the arguments that the CT/AT "omit" verses and then others assert that the MT "adds" verses. Common examples are the ending(s) of Mark and the Johanine Comma (I John 5:7-8). There is debate over this and both sides have interesting and valid points. I've gone back and forth, but I tend to favor the CT these days, especially with the Mark ending...the long ending just sounds very much like a summary that could have been added. However, I've seen some compelling arguments from the MT wing. I think the TR is solid and you can meet God through it, but I don't know that it's the "best." I am not an expert and generally defer to the scholars.
A lot of people get nervous with critics, and so you see attacks on the guys who began the criticism tradition (Westcott & Hort). They were occultists, but there have been a number of very good and very Christians scholars since who have verified things.
I do want to emphasize that the variations never change doctrine. I've seen this put forward a number of times, but its just not true. Often the examples are put forth in a vacuum when another verse (or verses) confirm things.
