Before we start, go grab your bentcher-of-choice. If you don't have a bencher at home (which you should), grab your siddur. Bentching is listed as "Grace After Meals" and can be found under the heading "Blessings" or similar. As a practical matter, I prefer the NCSY Ivrit bentcher because it has the transliteration in addition to the English translation. (There are both Ashkenazi and "Ivrit" editions with transliteration to match those Hebrew accents. Note that the Ashkenazi version uses the "oy" Ashkenazi accent, which may not match your community's Ashkenazi accent. In other words, take transliteration with a grain of salt, and try not to use it as a crutch.) I hate unfamiliar bentchers, but that's a different post.
A mezumen is the "quorum" required to say the formal "invitation" to bentch. Unlike a minyan (the "quorum" for prayer), a mezuman requires only three males over bar mitzvah age who have eaten together. The privilege of leading the mezumen is an honor. As a default, it goes to a kohen. The kohanim may decide between themselves who will lead, all other factors being equal. The kohen may defer to someone else, and his deferral is required for a non-kohen to lead. If there is no kohen, there are many options. [These are also the people a kohen may choose to defer to.] The honor is often given to "the guest" if only one male guest is present. If many male guests are present, it can go to the most senior Torah scholar, someone else learned, or someone celebrating (whether a formal simcha like a childbirth or something as simple as a new job).
If you are offered the honor, you may decline it. No questions asked. In all my meals, I have never seen anyone question why a person turned down leading the mezumen. And it's turned down pretty often! I tell you this so that you will know that refusing the honor will not "out" you as a conversion candidate (who should NOT lead) or as someone who is simply nervous or uncertain. Say "no, thank you," and the host will move on.
Most notably, the people not at "your meal" do not count for a mezuman. For instance, in a restaurant. (However, it is common practice to grab any "missing" men for a minyan from other tables in a restaurant for a sheva brachot meal after a wedding.)
If there are ten or more men present at the meal, the name of G-d is inserted into the mezumen. That is the word in parentheses ( ). If there is not a minyan present, do not say the words in parentheses.
There is a female version called a mezumenet. This is when three or more women over bat mitzvah age eat together. There is a difference between the rabbis whether the meal may have a mezumenet with less than 3 males present or whether they must be female-only. I have seen meals which have split into male and female groups in order to have both a mezumin and a mezumenet. Three or more women eating together without a man present can certainly make a mezumenet. However, my understanding is that it is voluntary. The women may bentch individually if they want to (or simply don't want to make a mezumenet).
The concept of the mezumin will most often come up on Shabbat. On Shabbat, the bentching will begin with the singing of Shir Hamaalot, the psalm at the very beginning of the printing of the bentching, possibly set aside in a darkened box (like the NCSY), smaller print, or some other method.
Then the leader will begin. You will say the portions marked as "response" or "the others answer," etc. This will be very fast. I personally find it to be one of the hardest things to memorize (I haven't) simply because it's so fast. Either find a transliteration or say it in English. No one reads Hebrew that quickly without it already being memorized. (Assuming you aren't a native Hebrew speaker, of course.)
The leader will also read certain parts louder than the rest. If you hear the leader say a blessing beginning "Baruch atah...," it is appropriate to answer "Amen" and then continue your private bentching. Almost everyone will respond Amen to the blessing "hazon et hakol," (if they don't, they simply may not have heard it) but most people don't seem to respond to any other part. There is also a problem with the leader mumbling his parts, so you may not understand where he is until you have bentching memorized much better.
The purpose of the leader reading the bentching out loud is a discussion for another day.