about:memory is the easiest-to-use tool for measuring memory usage in Mozilla code, and is the best place to start. It also lets you do other memory-related operations like trigger GC and CC, dump GC & CC logs, and dump DMD reports. about:memory is built on top of Firefox's memory reporting infrastructure.
BloatView prints per-class statistics on allocations and refcounts, and provides gross numbers on the amount of memory being leaked broken down by class. It is used as part of Mozilla's continuous integration testing.
Refcount tracing and balancing are ways to track down leaks caused by incorrect uses of reference counting. They are slow and not particular easy to use, and thus most suitable for use by expert developers.
Valgrind is a tool that detects various memory-related problems at runtime, including leaks. Valgrind is used as part of Mozilla's continuous integration testing, though the coverage is limited because Valgrind is slow.
LeakSanitizer (a.k.a. LSAN) is similar to Valgrind, but it runs faster because it uses static source code instrumentation. LSAN is part of Mozilla's continuous integration testing, with most tests running through it as part of the AddressSanitizer (a.k.a. ASAN) test jobs.
Apple provides some tools for Mac OS X that report similar problems to those reported by LSAN and Valgrind. The "leaks" tool is not recommended for use with SpiderMonkey or Firefox, because it gets confused by tagged pointers and thinks objects have leaked when they have not (see bug 390944).
LogAlloc is a tool that dumps a log of memory allocations in Gecko. That log can then be replayed against Firefox's default memory allocator independently or through another replace-malloc library, allowing the testing of other allocators under the exact same workload.