Aspects are the bridge between the crunch, the rules that provide the structure to the game, and the narrative.
They are both a symbol of some important element or essence of the game, and a tool that can be manipulated in a narrative sense or to add a real bonus or penalty to the math used to resolve conflict. You can think of them as icons that represent part of the operating system of the game. Or you can think of them as the quote from a larger speech. They might be pretty, or descriptive, or be lyric from a song, or just about anything, but they all represent a whole lot more than their literal meaning.
When Nathan Hale (purportedly) said, "I only regret that I have but one life to give for my country.", he meant it literally, but he also was using a sort of conceptual metaphor to explain to his British captors the importance of sovereignty to Americans, even in the face of death. He didn't regret dying for his cause, only that he wouldn't be able to help fight the Revolutionary War any further. Hale himself has become an allegory for continued struggle in the face of dire consequences. Using a portion of Hale's famous last words, something like, "...but one life to give..." would make an excellent character aspect. Narratively, it represents all kinds of powerful ideas of freedom, bravery in the face of death, the importance of self sacrifice, etc. (Thanks wikipedia!)
Aspects always have more than one meaning, and usually there is at least one positive and one negative interpretation. "...but one life to give..." might show that Hale is devoted to his cause even in the face of overwhelming odds, but it also might mean that he is less able to turn down incredibly risky assignments if they serve his country.
Characters aren't the only ones with Aspects. Every single important plot element has one or more. Important NPCs (including major villains), groups and organizations, religions, items, and locations all have Aspects that are either known or secret, and they can all be Invoked or Compelled the same way as character Aspects. A subway tunnel under the city might have the aspect "Inky Darkness", which the GM could invoke to give a bonus to the Stealth check of the gang that's sneaking up on you. An NPC the characters encounter could have the Aspect "Paranoid? More Like Realistic!" If the characters had discovered his aspect, they could use it to their advantage, playing on his paranoia to get the result they want from him. If they didn't know, the GM could decide they had triggered it by accident and all kinds of fun could take place.
Aspects have two main functions. They can either provide a direct mathematical bonus to a roll, or they can have a narrative effect. Say a hallway leading to the jewel exhibit at the Natural History Museum has the aspect "Sounds Carry". The characters run into a little scuffle with some guards who assume they are here to steal the museum valuables, and the GM decides that the (relatively quiet) grunts and impacts from the fight still echo down the hallway almost undiminished by the distance to alert the real thieves several rooms away, who make their escape.
The only effective way to explain exactly what Aspects are is by example. Lots and lots of examples. The only way to get good at creating Aspects is to do it.
There are however a few guidelines to making an effective aspect.
1) All aspects must represent something significant about your character.
Everyone gets a little angry sometimes, but somebody who is a "Powder-keg of Rage" will fly off the handle spectacularly. An aspect is something that is central to your character; strong personality traits, odd powers, events, titles, responsibilities. Likes Chinese Food is not a good aspect. It tells you something about the character, but nothing significant. "Broke His Leg When He Was Seven" is an even worse aspect. It tells you nothing significant AND it doesn't apply currently. However, if that broken leg came from falling from a tree, a good aspect from that would be Fear of Heights. I will use this aspect for my next example.
2) An aspect is more interesting if it is descriptive.
The more descriptive an aspect is, the easier it is to invoke or compel. You're showing you have an interest in this aspect, and want to see it in play. It's also more likely to catch your storytellers eye as something to use. Continuing from the previous entry, "Fear of Heights" is a fine, basic aspect. It's something that might come up on occasion, but you're not too invested in it. However, describing it as "I Stay on the Ground" implies that your character doesn't even like to be up a ladder or the like, though it could also imply a fear of flying or sailing. If you want to expand it like that, it would be a good choice as it could come up in a variety of situations, giving your Storyteller something to play around with. To represent a true fear of heights at the extreme, an option might be "I'd Rather Fight that Demon than Jump!" which states flat out that heights are scarier than monsters to you. If you take that, expect heights to be a big obstacle for you in the future.
3) You need 'good' and 'bad' aspects, preferably mixed bags.
Often, there is a temptation to make a character with only 'good' aspects, ones you can see using a lot. This is a trap. Aspects like that gobble up your fate points without generating any. Likewise, having only 'bad' aspects will generate fate points aplenty, but then you'll have trouble spending them. Mixed bags are aspects that can be good or bad depending on the situation. Things like "I got Luck, Good and Bad" or "Ghoul Hunter", which can generate and spend fate points. It's possible to go with all mixed bag aspects, but it can make your character seem a little generic. Not too good or too bad at anything. As such, a personal recommendation of mine is to have one 'good' aspect that you can regularly use that represents a strong point, a 'bad' aspect that generates fate that represents a weak point, and mixed bags for the rest, showing the areas where they are average. You are of course free to change the ratio if you want.