Okay, BRP stands for Basic RolePlaying, and is the line of games descended from RuneQuest. There's been a lot of evolutionary bifurcation along the way, so what I'm going to talk about doesn't apply to every offshoot (parts of it don't describe Mythras, for example), but it at least describes RQ1-3 (and I think, but don't know for sure the current version), the standalone BRP book and some others.
There are two methods of advancement in traditional BRP: learn-by-using and training.
The former operates by giving you a check mark (sometimes called a "tick") in a skill the first time you successfully use a skill in an adventure. Since BRP is a percentile based system, at the end of the adventure or whenever there's some downtime (these don't have to be the same thing) all skills with ticks are rolled again, usually with a bonus based on either your skill modifier or Intelligence depending on which particular incarnation of the rules you are using, and if you exceed the current skill value (or get a 96% or greater roll in any case) you advance (in the earliest versions by 5%, in later versions by 1D6%). You cannot get more than one tick in each skill per adventure or time between downtime.
Training is done by spending time and money, and varies more in how exactly its expressed: usually there's been a cap about how high you can train (often 75%) and often can only be done once between advancement from use in the field. Some versions also allowed training to improve attributes though that's usually a long and expensive process.
A few observations, primarily about the learn-by-doing after using the game system for a long period:
1. There are some problems with how skills are used and handled. There are skills that are important but only relatively rarely rolled for, while others are rolled for frequently. The former can include things like outdoor skills (Climbing, Jumping or Swimming) while the latter include things like perception and combat skills. This can produce some lopsided advancement that is not clearly the way you'd expect things to go. Some versions have done various patches on this (in at least some versions if you were doing extensive Riding you'd get a tick in that even though a roll was never called for.)
2. This approach can also breed for what we always called "tick hunting", sometimes to a pathological degree. This is when a player basically hunts for excuses to use a skill just to get a tick in it. One would think, since normally its assumed you only get a tick when some risk is involved, this would be self-limiting, but that doesn't seem to stop people; in pathological cases people will even do this with skills they don't actually usually have much reason to bother with, just out of the compulsion to get that roll.
3. The approach only kind of crudely represents how much the skill is actually used, since only the first success counts (this despite the first note).
4. Some people find bookkeeping the ticks tedious (of course, those people often would find other elements of RQ at least tedious, such as the fact a combat round can involve as many five die rolls).