'Levels' is how Academy City measures the ability of it's esper students after they are first introduced into the Power Curriculum Program, ranking them on a scale of 0 to 5. Level 0's are those with no psychic ability, ability that can't be measured in the yearly System Scans, or so little ability that is of merit or demerit. Level 5's are the most powerful espers in Academy City, and perhaps the world considering that Academy City is at the fore front of psychic research and evaluation. Understandably, Level 5 espers are very rare to the point that, in a city of 2.3 million people where about one million and one hundred and four thousand have any psychic ability there would only be a handful (maybe) of level 5 espers. These rankings are then entered into Academy City's Data Bank, a repository of information on espers, experiments, and other matters where it can be accessed by people with the proper authority.
It's important to note, however, that an esper's level does not determine their potential, only their current ability. In canon, it's a mark of achievement for the Power Curriculum Program that they were able to advance a Level 1 esper into Level 5, one of seven. This also means that, if an esper's abilities deteriorate, that their level may very well fall too.
A Level doesn't necessarily mean that an esper has increased in raw power though, it can also mean that an esper has become skilled enough for new uses of their ability. This example is intended to show how an ability might evolve as it's user went up in Level, using the character Katagawa Sakurako (a Teleporter) as the example.
At Level 1, Sakurako is capable of teleporting a single object that she can touch up to a certain distance away. At her current capability she's only capable of teleporting something around 70 to 75 pounds up to 65 to 70 feet away, as long as she's capable of seeing the end point where the object will end up. This does not account for the specifics of a situation such as how tired she may be or if adrenaline is pumping through her system.
At Level 2, Sakurako is capable of teleporting larger objects farther distances but no longer needs to rely on her sight to send the object to the 'end point', as long as she is aware of where the object is in relation to herself. This requires an intense sense of spatial awareness of a three-dimensional area, but is still incredibly useful. In addition, she can now teleport an object weighing up to 140 to 150 pounds up to 130 to 140 feet away as long as she can touch it.
Level 3 is when Sakurako's ability to teleport really begins to come into it's own. She's now capable of the concentration needed to teleport her own body to a location within her sight, unless the location is extremely close by. In addition, objects weighing up to 215 to 225 pounds can be teleported up to 200 to 210 feet away as long as she knows where the end point is in relation to herself and she is touching the object.
At Level 4, Sakurako's ability to teleport objects and herself begins to take on a fantastic turn. Now she is capable of teleporting an object (or, in theory, herself) into another object. Rather then end up destroying the object or being unable to perform the teleportation, the object she's teleporting will displace the surrounding material until it occupies the space Sakurako intends it to. In such a fashion she can teleport nails into the ground to pin an opponent or slice through cement and steel with a pane of regular glass. At this level, objects weighing in at 285 to 295 pounds can be teleported up to 265 to 275 feet away. She may also teleport herself at the same range (within sight) though she can now 'safely' teleport herself to locations she can't see up to 100 feet away. Though unless she has memorized the 'end point', she risks teleporting herself into an object, meaning shorter jumps are still safer.
At Level 5, there is theoretically no upper limit to the weight of an object she may teleport or how far away she could teleport them. In addition she could, also in theory, be able to teleport another person with enough effort. Sakurako's skill as Level 5 will be largely dependent on her own practice with the ability and frequency of use. She could even learn to teleport something without having to touch it at all!