(1) Personal computers are more "user friendly." This means you generally don't have to spend time reading obscure documentation, like this. (This is because they are designed to be understandable even by idiots, which puts them almost within reach of physicists.) Furthermore, since most impersonal computers have their own graphics monitors (which makes them "workstations"), much of the nice stuff on personal computers can also be done on them, if the friendly System Manager has set it up for you. (Our friendly System Manager, Ming, can handle anything on our computers.) Personal computers have a Graphical User Interface (GUI), which means you don't have to do everything by typing. To an increasingly large extent, this is also now true on mainframes, which usually now have X-terminals or other graphics monitors, which not only display things in a more convenient way (e.g., allowing you to view preprints with their equations directly on the screen), but also allow your input by essentially just pointing and clicking (using a "mouse"). Unfortunately, such things are not standard on mainframes (old habits die hard, but the people who have them die), so you have to know where to find them. (Ask the Manager for the latest.)
(2) Since you use a personal computer alone, you don't have to worry about somebody else crashing it, slowing it down, or updating it while you're trying to use it. (Of course, you may crash it, slow it down, or update it yourself, but you have more control as to when this happens.)
(3) Mainframe computers are sometimes faster, which may be important if you are doing some serious number crunching (using programs such as Maple). However, the latest personal computers (such as PowerMacs) are so fast they are usually used to replace mainframes.
(4) Since mainframes/servers allow many users at once, they can also handle dumb terminals. Personal computers can also be used as terminals to login to mainframes (what a disgusting thought), and files can be transferred between the two.
Macs are really Unix machines in disguise, so all machines are more or less the same (unless they're running MS Windows, which is more popular in the business world, since it matches their taste in clothing). Other Unix machines can be (almost) as convenient to use as the Macs once you have things set up right.
(1) The "Operating System (OS)" (Mac OS X, Unix/Linux, MS Windows, etc.), is the master program which runs the computer as a whole. Its basic purpose is to translate between the "machine language" that the computer understands and another language, the "shell", which most of the people using the computer understand, or would if they took the time to read this file. Operating systems also take care of various housekeeping that needs to be done automatically. (Some obsolete mainframes use another operating system, VMS, which I won't talk about here because it's too disgusting.)
(2) The "Graphical User Interface (GUI)" (Quartz/Aqua, Gnome/KDE/X Windows, Indigo Magic, etc.) allows one to largely avoid the shell. It runs on top of the shell, and mimics many of the features of the Mac OS (but not quite many enough to get sued, apparently). The main features are that you can run programs or open directories simply by double-clicking on their "icons" (small pictures representing files) with the mouse, and feed a file into a program by just clicking on the file icon and "dragging" it (moving the mouse while holding the mouse button down) onto the program icon. (Some people claim that experienced typists can do this faster by just typing in the shell. Some people also claim there was once something called a "typewriter" that worked even better than a computer.)
(3) "Text editing" programs are for writing: programs, papers, electronic mail messages, etc. Besides just typing, you can conveniently "cut and paste" (move text from one place to another): no more white-out, no scissors and tape, no photocopying. The main (I use this adjective in a somewhat condescending way) text editor in Unix (except Mac OS X) is "emacs": It is much more versatile than many others (e.g., it can be used on any type of terminal), and can be "customized." (You can modify the way it works to suit you better, or get somebody to do it for you. Unfortunately, I can't help you right now, because I'm busy editing this stupid file.) There is another stupid (this adjective is meant to be even more condescending) editor on Unix called "vi:" As opposed to emacs, which does more than anybody could possibly want, vi does less than anybody could possibly want. If you're using vi, it's not my fault. One particularly useful feature of emacs is "2-window mode," which allows you to look at 2 different files at the same time: You can use one just for reference, or move text from one to the other. (The most useful feature of vi is that it continuously beeps at you.) Fortunately, there are now more Mac-like editors on any OS. (Yes, I do use a Mac, how did you know?) Some have 3-letter names that sound like interjections, like "zip", "jot", and "axe". (Holy cow, Batman!) Their advantages are that you can select the text you want to copy, cut, or delete by just clicking and dragging (with the mouse, please), and select where you want to put it by just clicking. (More recent versions of emacs do similar things.)
(4) "Typesetting" programs turn text files into files which can be printed with a quality much better than any typewriter, and even as good as professional books and magazines. For example, all your right margins will be perfect, since the program shrinks and stretches the spaces between words to make the margins exactly right. Also, you can easily change type styles, and there are literally hundreds of them (including sizes from 5pt upto 24pt and beyond). In particular, equations come out nice, including subscripts and superscripts. The only such program worth considering now is TeX. However, word processors are already close in power (and surpass TeX in convenience), and will replace TeX eventually.
(5) "Networking" means communicating with other computers: logging in, exchanging files (like preprints), electronic mail. The main network is the Internet. Networks let you send and receive mail, login to other computers, and have live (but often slow) 2-way communication.
(6) Programs for Mathematics include Maple, Mathematica, and Macsyma. Maple is the Major program used here. The rest are more specialized, except for Mathematica, which is more expensive. More general "programming languages" used mostly for doing mathematics include C (now standard) and Fortran (for old die-hards: see above).
Now that you understand the lay of the land, let's review what you should already know to have gotten this far, except that the guy standing over your shoulder giving you instructions talks too fast and you didn't get a chance to write it down. The first things you have to know are how to start and stop. (I'll assume you know how to turn the terminal on. Chances are it's left on all the time anyway.) To "login," when prompted by the computer with "login:," you type your "username," followed by a Return. (The Return key is also sometimes called "Carriage Return," or "CR," in memory of the times when the keyboard was part of a typewriter, whatever that is. Actually, some computers now have the advanced feature of calling this key "Enter". This is preferable, since it makes it look more like a calculator, which is more like what you're supposed to use it for.) If you have a dumb terminal, you may have to hit a few CR's before you even get the login prompt. (I guess they should call them "deaf-and-dumb" terminals.) Your username can be almost anything you like, so if you don't like yours, ask the Manager (or whatever they call themselves) to change it. (Usually it's your first or last name.) You are then prompted "Password:," to which you respond with your secret password (followed by a CR). "Secret" means it is not your first or last name (or anybody else's, or a real word, or one of those backwards), but something that nobody would ever guess, like "42ytw89fej". On Unix, as soon as you login, the computer runs the program files ".cshrc" (or maybe ".tcshrc", or ".xsession", or even ".4Dwmrc"; I hate Unix) and ".login" (note the periods at the beginning of the names), which set things up for you: For example, they may tell you the date and time, or whether you have any mail. They may also contain some personal definitions which make using the Unix operating system easier for you.
If you use a graphics terminal, logging out may be easy: This depends on your setup, but you can arrange to be able to logout by just pressing the right mouse button in the right place (or maybe it's the left mouse button in the left place). Otherwise, things may be more complicated. On a dumb terminal, to "logout", you type either "logout" (and CR), or else "^D" (no CR). ^D, pronounced "control-D," means to hold down the "Control" key while hitting the "d" (no shifting necessary). The computer considers such a "control character" as a funny letter in an extended 128-character ("7-bit") alphabet (which includes the usual 26 letters in both lower and upper cases, the 10 digits, space, everything else on the keyboard, ^A through ^Z, and maybe some other garbage). You can think of the Control key as a different kind of Shift key. (Actually, CR is the same as ^M.) Depending on what's in your .cshrc and .login files, you may be forced to type "logout (CR)" rather than just "^D." Also, you may need to do it twice.
A minor point: That moving blob or underline on your screen that follows your typing is called the text "cursor" (and you thought you were the curser). That's pretty obvious now, but more useful when you're editing text, when the place you're typing isn't at the end. If you are using a graphics monitor, there is also a mouse cursor, which moves as you move the mouse. It is used to select windows and text, click buttons in windows, rearrange windows, and lots of other fun stuff.