Levels or no levels... I personally don't mind levels even though I prefer systems where character's experience is determined otherwise. Two most common systems determining character's power and achievements are level based advancement and experience point based advancement.
Level Based Advancement
Usually you gain experience points what stack. When you have enough of them, you step up to next level. After leveling up, you assign new points and powers to your character, what are usually pre-determined based on the new level achieved.
Usually you cannot develop your character between the levels. The reward of experience comes stacked. After assigning the new points and benefices, you start collecting experience points for the next needed amount of experience to gain new level.
This is really simple in the cost of free customization of the character. You get certain packet of new things, like everyone else does. You might assign some points in pre-ordered places, some you might spend where ever you want to.
Advantages are, that it is easy to see how powerful the character is. Just check his level, and you have a idea of characters capability, so it is easy to design quests and adventures.
Also quick resolution of new powers after leveling up is relatively fast. You know what you get and where to put those extra points and powers.
There are also disadvantages, for example no free customization or linear advancement.
Experience Points Based Advancement
Other is, where experience points you gain can be converted any time to points to spend on character advancement. There might be some restrictions based on character, or some attributes and stats might be more expensive than others and usually more higher stats get, more experience they need to be spent to rise further.
This system is really versatile and you are quite free to advance your character to the direction you want to. Also it is non-linear, so you don't need to wait for next milestone to advance your character.
Problems are, that it's hard to determine different characters' power levels, as there is no guide how to compare them. You can write down total spent experience, but still there are differences how those points are spent. Both characters with exactly same amount of spent experience might be totally different. Another might have spent all his experience points to powerful but specific skill, when another has got much more attributes with lower cost. So which one is better?
Also as there is no guideline of power levels, it's slightly harder to make quests not too hard, but not too easy either.
How To Get Experience Points
Well, in both cases doing stuff. Usually level based experience systems have more strict guidelines how, when and how much to award experience points based on tasks, kills, loot etc. when free experience point systems give guidelines based on what happened in one session. Differences are easy to measure with an example:
Level based system:
Killed 10 goblins á 25 EXP
Got 1000 Gold loot á 0,1 EXP
Reported successful quest 250 EXP
(Random bonuses from cool stuff or achievements = ? EXP)
= Total 600 EXP
Free experience points system:
Experience after every session 1 EXP
How well did player roleplay his character? 1 EXP
Did player character or group achieve or advance in adventure? 1-3 EXP
Was player in danger and survived it? 1-2 EXP
= 1 to 7 EXP depending on this and that
What Do I Prefer
I like free character development of experience converted to character points systems. You can make your character what you want to. Problem is, that usually I just forget to give EXP after sessions, and otherwise give them randomly. It's easier to just say first experience points value you come up than start to think what happened when you don't have got a real guideline.
In level based guidelines are great and the fact you know how powerful your (or player's if you're GM) character is. Problem is, that even the experience based changes can be small in exp to pts development, you might need to wait longer to level up without any advancement.
Both systems get thumbs up from me, even though either of them are not flawless. Still, if I have to choose between them, I think level based experience looses.