After a successful "make", try "make check".
It's possible to run test cases individually, and get verbose output when one fails, etc. For more information, see the "Running test programs" section of HACKING.
Test programs live in tests/. They mostly use a standard test harness, in tests/harness/, which wraps each test, reports results, and generally packages things up nicely. The test harness counts how many testcases pass/fail/skip, catches signals and unhandled exceptions, and so forth. It can also also check for memory leaks and accesses to uninitialised values by making use of valgrind, for platforms which valgrind supports (configure automatically enables use of valgrind if a suitably recent version is detected).
A typical test program has three parts: the tests themselves (at the top), a table of tests (at the bottom), and a tiny main which sets the test harness in motion. It uses the table to figure out what the tests are called, and what function to call to run them.
The most important test system for most people will be apitest. This also uses the test harness, but has several tables of tests to be run depending what facilities each backend supports. A lot of the work is done by macros and helper functions, which may make it hard to work out quite what is going on, but make life easier once you've grasped what's going on. The main() function and other bits are in apitest.cc, and tests themselves are in various other C++ files starting api_. Each one of these has its own tables for various different groups of tests (eg: api_db.cc, which performs tests on the API that require a database backend, has basic tests, a few specialised groups that only contain one or two tests, tests that require a writable database, tests that require a local database, and finally tests that require a remote database).
To add a new api test, figure out what the test will be dependent on and put it in the appropriate place (eg: if adding a test for a bug that occurs while writing to a database, you want a writable database, so you add a test to api_db.cc and reference it in the writabledb_tests table).
Currently, there's api_nodb.cc (no db required, largely testing query construction and boundary conditions), api_posdb.cc (db with positional information required) and api_db.cc (everything else, with lots of subgroups of tests). It's easiest to base a test on an existing one.
You'll notice in apitest.cc that it runs all appropriate test groups against each backend that is being built. The backends are inmemory, multi, chert, glass, remoteprog and remotetcp. If you need to create a new test group with different requirements to any current ones, put it in the appropriate api_ file (or create a new one, and add it into Makefile.am) and remember to add the group to all pertinent backends in apitest.cc.
Incidentally, when fixing bugs, it's often better to write the test before fixing the bug. Firstly, it's easier to assure yourself that the bug is (a) genuine, and (b) fixed, because you see the test go from fail to pass (though sometimes you don't get the testcase quite right, so this isn't doesn't always work as well as it should). Secondly you're more likely to write the test carefully, because once you've fixed something there's often a feeling that you should commit it for the good of the world, which tends to distract you.
The framework is done for you, so you don't need to worry about that much. You are responsible for doing two things:
Adding the test is simple. There's a test_desc array in each file that comprises a set of tests (I'll come to that in a minute), and you just add another entry. The entry is an array consisting of a name for the test and a pointer to the function that is the test. Easy. The procedure is even simpler for apitest tests - there you just use DEFINE_TESTCASE to define your new testcase, and a script picks it up and makes sure it is run.
Look at the bottom of tests/stemtest.cc for the test_desc array. Now look up about 20 lines to where the test functions are defined. You need to write a function like these which will return true or false depending on whether it failed or not.
In addition, there are a bunch of macros to help you perform standards testing tasks. Things like TEST_EQUAL are all in tests/harness/testsuite.h. They're pretty simple to use.