This question is sort of 3-fold:

(1)What are the problems/issues that database clustering seeks to address?

(2)What is database clustering and what are the current approaches,
solutions, and products (in other words, how are the problems being
addressed and who is addressing what problems - I know a couple of the
main products that are being used for DB clustering are MS SQL server
and Linux-based mySQL server)?

(3)Finally, what improvements could be or need to be made in database
clustering (maybe what improvements are being suggested)?

Comments: This last question is especially important.  If you have to,
spend less time explaining the first 2 (more of (but not exclusively)
-pointing- me to information rather than explaining it yourself).  Also,
looking at some of the previous questions you answered, I know you
guys are generally pretty quick (*real* quick sometimes!), but I just
wanted to make sure you could get a response by, say, later this
afternoon or evening - later tonight (9 or 10 or 11ish) at the very
latest (today being Tuesday, November 30).  Thanks, guys!

There are 2 main reasons:
1. scalability
2. Reliability (failover)

You could look at Oracle's RAC (real application Cluster's or their
Grid concepts)  www.oracle.com or otn.oracle.com (technical side). 
From a high level they use a shared disk approuch (each machine in the
cluster can see the same disk) and a high speed interconnect to manage
the cache between machines.

Some vendors (eg MS Sql Server, IBM DB2) use a shared nothing
approuch.  Here the data is actually split up onto different machines
and then the different machines can get you the informaion you want. 
Of course, if you add another machine into the mix you have to
redistribute the data.

Each approuch has its advantages and disadvantages and each vendor
will tell you theirs is best.  But they should have plenty of white
papers describing their clustering solutions and it is enlightenening
to at least read them.