I'm new to photography, just trying to get the right lens!
It is compatible and you can use it (the 600D can use both Canon EF and EF-S lenses), but whether or not it's the "right" lens depends on a lot of other factors: what/how you want to use the lens and your personal budget being two of them.
My first advice is that 50mm may be a bit "long" on a crop-body camera like the 600D. This is a matter of personal taste, but if you have an 18-55 kit lens, consider that a 50mm lens, is pretty much like being zoomed all the way in on an 18-55 all the time. If you want something a bit wider for walkaround usage, you may want to consider the EF-S 24mm f/2.8 STM--it's slower (has a smaller f/2.8 maximum aperture), but smaller and wider, and the cost is roughly the same.
Secondly, the 50mm f/1.8 II is an older lens design hailing from the late '80s. The current version is the EF 50mm f/1.8 STM. Most of the improvements are on usability, rather than image quality, but most folks would prefer the newer version, particularly if they planned to shoot video with it.
Thirdly, if you need something cheaper, there is an alternative that actually exists (costs around US$50), but you may have to compromise on image/build quality. And that would be the Yongnuo 50mm f/1.8. They also offer a 35/2 for about US$90.
All DSLR Canon EOS Bodies are able to mount EF lenses and APS-C Bodies are also able to use EF-S lens. Only the EOS M (mirror-less) series is not able to mount those lenses without an adapter.
A more details could be found at What do all those cryptic number and letter codes in a lens name mean?