Good answers, and thanks to those who gave them. I'll provide a bit more detail, since Max is trying to fill in the blanks on his character sheet.
BRA, Best Relevant Attribute, as mentioned, is the one attribute in a bias area that you use for skills in that area when you do them. Each bias area has three relevant attributes; it's assumed that whichever one is your best is the one you use in that bias area. Thus for tech it's intellect, intuition, and education level, so whether you are learning or doing a skill, one of the factors is whichever of those is the best for you. Intellect means you do it because you're that smart, intuition because you're good at figuring things out by the seat of your pants, and education level because you've picked up so much information about everything that you know something about this. In psionics, it's persuasion, intuition, and will power; in magic, animal magnetism, intellect, and intuition; in body, strength, agility, and will power. Note that for each bias area you use your best of those three attributes, even if you don't see how it works--like, what difference does strength make to smelling something? This keeps the playing field more level and simplifies the game, since you're not searching for the right attribute for each skill.
The bias of the highest skill you have in any bias area is your bias level, with a caveat: @0 skills do not count. Thus if you have a P0@1 teaching skill and a P8@0 posturing skill, your bias is 0@1, not 8@0. This is because the @0 skills don't really use the essence of the bias area. Another good example is in technology, where your general science skill (and many other sciences, such as computer theory) is an @0 skill--it doesn't actually mean you can do anything, only that you understand the theory behind it.
SAL is skill ability level, how good you are at doing something. On the On the Fly sheets, it's the number in the left column; the number in the right column associated with the bias area name is the bias of the skill.
You never roll to improve a skill, specifically. You do roll to use the skill in a new situation or a new way, and if it is clearly a new use of the skill and you use it successfully, you get a "skill mark" or "mark" on that skill. When you have amateur level skills, a skill mark is an increase of one intensity; at professional level, an intensity costs four skill marks, and at expert level it costs twenty-seven skill marks (experts do not really improve very quickly by thinking of new ways to use their skills). You tell the referee if you think what you are doing constitutes a new use; he decides if you are right, depending on how different he feels it is from what you've done with it before, combined with how many different things you actually could do with it. (New use for 180 degree skid turns is going to be a lot more generous in its definition than new use for driving motor vehicle, particularly if the latter includes doing 180 degree skid turns as a new use.)
You also improve a skill by practice. The rule of thumb in the book is that it takes sixty hours of specific practice over one hundred days to improve a skill by one mark; you can't on your own shorten the number of days, although if you intensify the program you can earn multiple marks for improving upper level SALs (a "double doubled" program can earn you four marks in one hundred days, advancing a level two intensity; you can double the number of hours spent practicing, and any other aspect of what you are doing, or you can quadruple the number of hours). If you have a skilled teacher or trainer overseeing your practice efforts, that automatically cuts your time (both hours and days) in half, and if that teacher/trainer makes a successful roll on his teaching/training skill, that cuts the times in half again.
The referee has the discretion to determine that your ordinary activities in a particular world or situation constitute ongoing practice of a skill. For example, John Cross is currently working as a blacksmith. He's not said anything about practicing his blacksmithing skills, but after a hundred days he'll improve that because he has been doing it every day. In the Mary Piper scenarios, there is a skill and attribute improvement track for each job description on the ship, such that just by being in security you improve your weapons skills, or by being in medical you improve your medical skills.
It's a bit different with attributes.
Generally if you decide you want to work on improving your skills, you write up a "training program" that describes what you're going to practice when, and commit your character to spending that time, and I tell you whether that is going to be sufficient to improve your skills and attributes as desired. Usually three hours in five days will get you advancement in about three months on skills, at amateur level.
I keep thinking of threads I should create and make sticky, but I don't usually get to them, and I don't really want to clutter up the top of the board with stickies, so I always err on the side of neglect.
--M. J. Young
How do I figure out when to roll for improving a skill, and what would the chances of success be?