A thread dump is a dump of the stacks of all live threads. Thus useful for analysing what an app is up to at some point in time, and if done at intervals handy in diagnosing some kinds of 'execution' problems (e.g. thread deadlock).
A heap dump is a dump of the state of the Java heap memory. Thus useful for analysing what use of memory an app is making at some point in time so handy in diagnosing some memory issues, and if done at intervals handy in diagnosing memory leaks.
This is what they are in 'raw' terms, and could be provided in many ways. In general used to describe dumped files from JVMs and app servers, and in this form they are a low level tool. Useful if for some reason you can't get anything else, but you will find life easier using decent profiling tool to get similar but easier to dissect info.
With respect to WebSphere a javacore file is a thread dump, albeit with a lot of other info such as locks and loaded classes and some limited memory usage info, and a PHD file is a heap dump.
If you want to read a javacore file you can do so by hand, but there is an IBM tool (BM Thread and Monitor Dump Analyzer) which makes it simpler. If you want to read a heap dump file you need one of many IBM tools: MDD4J or Heap Analyzer.