Relation between type-checking decidability, typability decidability and strong normalization
I'll give a more focused and technical answer to Martin's. If we're interested in dependent type theories, then neither direction is obvious, but both are generally assumed to hold. However the direction Decidable $\Rightarrow$ Normalizing is open, I believe, even for relatively well understood systems, though we do have partial results. This question explores some of the same issues.

More technically: a good technical framework for these questions is the setting of Pure Type Systems in which normalization does imply decidability of type checking. The result is folklore, but non-trivial, since a strategy for when to apply the conversion rule needs to be found. The obvious strategy is to normalize inferred types before all applications of the application rule. There are some subtleties even here, where the correctness of a natural (more efficient) strategy called expansion postponment was open for a while, but has been settled since.

There are type systems which are normalizing but undecidable, most notably the system with intersection types which types exactly the normalizing terms, and so must be undecidable (since otherwise it could be used to identify normalizing terms). It would be tempting to say here that the system is defective in the sense that it is not type directed, i.e. the terms do not contain enough information to figure out the type.

Now the converse, which would suggest non-normalizing systems are undecidable, is far from obvious: it was even open for a while for the famous $Type:Type$ system of Martin-Löf! This case was solved by giving an example of a looping combinator, which allows writing arbitrary computations.

However the general problem is still open for general pure type systems, though it is widely believed to be true, and some partial results have been shown in this direction, e.g. here.