Levels are a description of how much responsibility, initiative, expectations, and pay you receive. These are present at every (tech) company, even though they are named slightly differently depending on what company you're at. See the next section for more details.
The lowest level possible if you're a technical intern is going to be L59 - no particular reason that we know of as to why though 😉
Check levels.fyi to see how the Microsoft levels match up with levels at other companies such as Google, Facebook, and Amazon. Be warned, some of the numbers there don't quite match up.
This is the "intro" role, essentially the new hire role. If you're SWE, this is SWE 1. If you're DS, it's DS 1, etc. Your expectations are dependent on the role, but it's that you can contribute to a team, work independently if need be, and meet your deadlines. Basic stuff.
L60 requires a little more motivation, a little more initiative, and perhaps owning the project a little more. At this point you've graduated from the "trial" period, and are expected to perform like it.
This can be an intro role, but almost never is for bachelor's students and rarely is for master's students. PhD students regularly get 61 and frequently get 62.
This role expects you to take ownership of your projects and tasks. Don't just wait to be given tickets, be the person in the meeting with your manager writing the tickets for your project. Figure out who needs to be contacted, what customers you need to get involved with for early feedback, what other help you might need, and what kind of time frame you need. These two roles are all about showing that you don't just crank out code, but that you can actually get involved and really push for the right things to happen to get your project done on time. Note that sometimes this means letting your manager know that the timeframes they set aren't realistic, or that they give way too much time for something small. It certainly does not mean being a "yes man"
62: The same as 61, but more pronounced. More ownership, potentially of multiple small projects.
Nowhere on record do we have this being an intro role, but new hires with a few years of prior experience can get it.
This role expects you to be able to manage people as well as projects. You should be able not only to manage "borrowed" developers for your projects, but also developers assigned directly to your projects. Part of this involves figuring out how many you need, and what kind.
You should be able to help onboard people (mentioned specifically in some internal doc I read at some point). At this point, there should be no project you don't want to take ownership of, at the very least owning your section. Constantly be trying to learn more, and you should become a resource for the L59-L62s.
64: Largely the same, but larger projects, more focus on ownership, and of course higher performance expectations