Disclaimer: I am not an academic. The following post is akin to a plumber's guide to brain surgery.
Type inference is complex and has evolved over time. In this post, I will try to explain how I see the landscape and where LHC and other Haskell compilers fit into this landscape.
The beginning: The Hindley–Milner type system. 1982.
The typing rules here are quite simple and every Haskeller seem to learn them intuitively. They include things like: if 'f :: A → B' and 'a :: A' then 'f a :: B'.
In this system, types are always inferred and there must always be a single, most general type for every expression. This becomes a problem when we want higher ranked types because here a single, most general type cannot be inferred. There may be many equally valid type solutions and it has to be up to the programmer to select the appropriate one. But this cannot happen in plain HM as type signatures are only used to make inferred types less general (eg. [a] was inferred but the programmer wanted [Int]).
Omitting the type signature in the following code can show us what plain HM would be like:
In GHC, the snippet will run fine with the type signature but not without it.
Version two: Bidirectional type system. 2000.
People realised that checking the correctness of a given type is much easier than inferring a correct type. Armed with this knowledge, a new type checker was born that had two modes usually called 'up' and 'down'. The 'up' mode lifts a new correct type up from an expression and the 'down' mode that checks the correctness of a type. Because of these two modes, this kind of system was called bidirectional and it deals with higher ranked types quite well.
LHC current implements this.
Version three: Boxy types. 2006.
At this point it had become apparent that higher ranked types didn't really play well with higher order functions. People often found themselves in situations where slight, seemingly innocent changes caused the type-checker to reject their programs. An example of this can be seen in this gist:
Impredicative polymorphism is required for the above code and boxy types is a stab in that direction. Bidirectional type checking was a big improvement over plain HM but it lacked granularity. Types are either 100% inferred or 100% checked with no middle ground. What if you wanted to check parts of a type and infer the rest? Well, boxy types solves exactly that problem. Boxes are added (internally, we're not making changes to Haskell here) to types and they signify an unknown that should be inferred. Now parts of types can be checked while the boxes are inferred and we're left with the best of both worlds. This is what JHC implements, btw. Boxy types was also implemented in GHC but was deemed to be too complicated.
Version four: FPH, First-class Polymorphism for Haskell. 2008.
Impredicative polymorphism, second attempt from the authors of boxy types. Improvements were made but the problem is still not solved.
Version five: OutsideIn(X). 2011.
GHC is a hotbed for experimentation in type checkers. GADTs, multi-parameter type classes, type families. These are just some of the features that makes the type-checker the largest and most complicated component of GHC. To deal with all of this, researchers came up with OutsideIn, described in a paper longer than all the previous papers put together. The algorithm is relatively simple, but, for practical reasons, implementations must reject some programs that are valid according to the specification.