Hi
Well, where to start...?
A Moby is a stretchy wrap - there are other brands of similar products. I think they're all essentially the same, but with small differences in thickness and width of the material (and price!) You can even make your own
These are great for newborns, easier to use than a woven wrap (non-stretchy) as you can tie first and slip bub in and out after. They are not so great as bub gets heavier, however, as the fabric stretches and sags. They're usualyl recommended to 8kg, but different parents will find them useful for shorter or longer, depending o ntheir needs. You can't back carry in a stretchy wrap (generally speaking).

An in between option for stretchy wraps is the Caboo (formerly known as close carrier). It's more structured, which means it may be easier to use, however this also means it may not fit everyone comfortably.

Woven wraps - there are thousands of different ones, lots of different brands, lots of different weaves, fabric blends, patterns and colours.... it's hard to know where to start. There are some more reaonsably priced 100% cotton ones, which are generally recommended to start.
These you have to tie around your baby, but can be used for front, hip or back carries. Depending on the wrap, they can also be comfortable right through the toddler years, so may last you longer.
The downside to wrapping is that it does take a bit of time to get the hang of it and it can seem too fiddly for some. IT's worth it, in my opinion, for the most comfortable carry - totally adjustable, you see, to fit you and your baby like a glove.

Slings generally refer to the one-shoulder carriers - can be ring slings or pouches (steer well clear of bag slings, which are dangerous). Ring slings are totally adjustable, whereas pouch slings are sized (though soem come with rings for minimal adjustment). It can be a bit harder to get your baby in safely and comfortably in a pouch sling, but it's usually ok if you get instructinos and the right size.
The main differences with ring slings is the material used - which can make a big difference to comfort (and beauty) - and the type of shoulder that it has (different people prefer different shoulders, I've never really used a ring sling so can't comment from personal experience).

Other options for more structured carriers, include the soft structured carrier (SSC) like an ergo/beco/manduca/olives&applesauce, etc, etc, etc (heaps)
Or a mei tai, which is essentially a square of fabric with straps that tie around the waist and shoulders.