Heading off to sleep now, but i'll try and give you a quick overview.
Your XP is broken down into three parts listed under DEVELOPMENT - PILOT, COMBAT, OTHER.
Initially you will tend to use them all up. You will max out your piloting skills using the first set of XP figures (PILOT), Combat using the second (COMBAT) and systems/social/survival using the third - OTHER.
However, you will hit a maximum that you cannot pass (you see the current maximum per skill under the MAX column for each skill).
This maximum is set by your attributes (Charisma, Reaction, etc). The only way to increase this maximum is to spend XP in ALL three categories (PILOT, COMBAT, OTHER) to up your attributes (example DEXTERITY, INTUITION, etc). The higher you want to increase this, the more XP you need in all three categories - so you will then have to save XP from one mission to the next.
The amount of XP you need to spend in all three categories to increase an attribute is shown under the DP column beside each attribute.
During the mission, the more movement/jumps/checks you pass will increase your PILOT XP. COMBAT XP is increased by.. well.. combat :-) OTHER is increased when you pass checks in survival, make good scanner locks, damage repair attempts successful.. etc.
That's a general overview.. other players here can get more specific. The manual (or guides in docs folder) explains the actual figures better and the relationship between the different attributes/skills and the different XP types.