A lot of people in this thread are using too much real science for my tastes. :) So here's a "for beginners" version.
Jumpships are the vessels that actually make travel between stars possible. They are long, thin, fragile ships that contain the Kearny-Fuschida (or KF) drive that let's them "jump" between stars, up to 30 light years at a time. Jumpships are expensive, hard to make, easy to damage, and for most of Battletech's history there are never enough of them. As a result, attacking a jumpship can be considered a war crime, depending on what historical period you're in. For their spacewarp jump drive to function, they have to be very far from any gravitational sources. As a result, most jumpships appear way "up" above a solar system, or way "down" below it. These are the zenith and nadir points. They are easy for crews to calculate, and there's so much room there (think of how much space is available "above" the solar system) that ships don't run into each other. The conventional drives on jumpships are really low power, so it would take months and months to travel from a zenith point all the way to a planet. That's why jumpships have docking collars so they can carry...
Dropships are big, robust ships that carry cargo. They can be heavily armed with weapons and armor, or they can be civilian haulers. Dropships all carry massive fusion thrusters to propel them from the jumpships towards the planet. Because they don't have Star Trek engines, it takes dropships about a week to two weeks to travel from the jumpship to the planet. They accelerate at a 1G (9.8 meters per second per second), like you do if you're falling, constantly gaining speed as they move towards the planet. At the halfway point, they turn around and use the same engines to start slowing down. Dropships can land on planets and take off again. They carry food, equipment, soldiers, etc, and make interstellar commerce (and conquest) possible.
From my recollection, jumpships are about a half mile long, and carry between 2 and 9 dropships (depending on what type of jumpship, and what type of dropship). Dropships can be about the size of a football stadium. So one jumpship actually carries a lot of stuff.
Warships are massive jumpships that have been built ultra-big and ultra-durable for combat. For most of the timeline, all the warships have been destroyed and can't be rebuilt. Later points in the setting bring warships back, but they've never really had a good set of game rules that were balanced, and so they're generally a "background only" element of the game. They don't have force fields or anything like that, just massive plates of armor. Some of them are durable enough to tank nukes.
Pirate points are locations within a solar system where the gravity of the star, and all the planets and moons, have canceled each other out. But planets and moons tend to move around, so pirate points aren't stable. You'd hate to appear at a location close to your destination world (really close, so you can catch them by surprise or something), only to find that by the time your jumpship has recharged (takes about a week to 10 days), the gravity well has shifted and it's now too intense for you to jump back out. That would suck, so use pirate points carefully. Also, it's a lot harder to calculate them than it is normal jump points, so make sure you know what you're doing.
Command circuits are what kings and queens use to get around. Pay enough money, and you set up a "pony express" type handoff system where one jumpship takes you from system A to system B virtually instantaneously. Then your luxury shuttle transfers to a different jumpship that's all charged up and ready to go. 5 minutes later, it takes you from system B to system C. Then you move to a third jumpship, etc, etc, etc. With enough money, you can cross the entire Inner Sphere in a day (while with conventional travel it would probably take more than a year to go that far).
ComStar runs the Hyper Pulse Generator network. ComStar used to be the old AT&T of the Star League, back when everybody was unified and peaceful. Now they're a crazy cult who plot to seize power from the various kingdoms that make up the Inner Sphere. They spy on everybody, and for a long time they're "Mwah-hah-hah" evil, complete with rubbing their hands together and saying "just as planned..." But they really just manage to keep the status quo as it is. HPG systems use a process very similar to a jump drive, but these are massive systems located on planets, and since they don't actually send ships through (just data), they can operate within a gravity well. ComStar keeps a huge network, and charges out the ying-yang to send these messages.
Smart people don't trust ComStar, and while it's never been proven that they're bad guys, people in high places suspect. So really REALLY secret information is often delivered by courier. Of course, there are dumb people in the Great Houses too, who send classified info over the HPG network. ComStar says they don't read any of it, but (guess what) they are lying.
