In short, it means modern Hebrew classes. But it's more than that. It's a particular style of teaching a foreign language through intense immersion and often with a class that may not share any common language other than the one being taught. It almost always refers specifically to teaching Hebrew, but other languages can be taught ulpan-style. As a linguist, the ulpan method is as fascinating to learn about as it is frustrating to experience. Ulpan is being thrown from the frying pan into the fire. It's as close as an adult can get to learning a language like a child does, in my opinion.
From Day 1, you will ideally be using only Hebrew (with lots of pantomime and hand gestures), though that depends somewhat on the dedication of the teacher and the persuasive skills of the students who want the teacher to speak in English. Depending on the teacher, you may not learn grammer very well, but that's not really the goal. The goal is to get you speaking ASAP: at the bank, at the grocery store, at your kid's parent-teacher conference. It's the most practical approach to language learning you can find.
When taken as part of immigrating to Israel, there are often cultural components to the education, including holidays, etiquette, and slang. It's not just language; it's acculturation. Some have called it brainwashing, to be fair.
Traditionally, ulpan is a common experience that ties together generations of people who made aliyah to Israel. It's a rite of passage that goes back before the country was founded. Many new immigrants, especially those from poorer countries and those rescued from hostile governments, often spend the first several months in an immigrant absorption center, where they spend several hours a day in ulpan. As I understand, attending ulpan is totally voluntary (as is staying in an absorption center), however most olim appear to take advantage of it, especially since the first 5 months (currently) are free. I can't find it on the new Nefesh B'Nefesh website, but I know you used to be able to take an extra 6 months of ulpan at a very reduced rate.
But five hours a day, 5 days a week, of Hebrew for five months? Your brain might melt. Talk about a bonding experience.
Ulpan outside Israel is often closer to the traditional foreign language class you're used to. For example, in my experience, I attended class once a week for 2.5 hours at a time as a near-total beginner. Homework was assigned between classes, but much of the experience depends on your classmates and their dedication to the immersion method. Learning any foreign language requires a certain degree of willingness to look like an idiot, but ulpan will test your self-esteem's limits. One major advantage to learning in Israel with olim is that everyone already feels like an idiot in public and is therefore highly motivated to improve. Also, it's unlikely that your class can get too off-topic when there is no common language other than Hebrew. The problem is finding ulpanim programs outside Israel. Even in NYC and Los Angeles, that's no easy task. There is Rosetta Stone (which I enjoy and find tracks my ulpan curriculum really well), and there are online teachers available. However, I've heard mixed results from online tutors and tutoring companies. Do your due diligence before hiring a Hebrew teacher!
You can find more about the history and methodology of the ulpan experience available from the Israel Ministry of Foreign Affairs.
Looking at the larger picture, do you think learning modern Hebrew should be considered a necessary (or even important) part of the convert/BT experience, or should we be focusing our energy on earlier Hebrew and Aramaic for text work? I'm having a hard time answering that question for myself, so I'm curious what all of you think and what your experience has been!