They are a gold mine of information. It is as simple as that. Yes, yes there are lots of doctors' reports which often don't contain genealogically important information, but dig into the depositions and other documents and you generally won't be sorry.
The pension files I have seen (fourteen of them) fall into three categories:
Invalid pensions: for men injured during their service.
Widows' pensions: for women whose husbands either died in service or were injured during their service and died later.
Mothers' pensions: for mothers of soldiers who died in service. In these cases the mother relied upon the soldier at least in part for support.
While two of the files I have examined have been disappointingly short (twenty pages or fewer) most have been seventy-five or more pages. Buried in these files you may discover birth, death and marriage records. Even the doctors' statements may prove useful in explaining your relatives ailments as I recently found (Medical History Revelation). Beyond these things you will usually find much more. You'll gain insight into your claimant's life that you might find nowhere else. Among other information I have found:
1. The name of a soldier's deceased father that I had not found anywhere else.
2. Confirmation of the identity and marriage of my ancestor's sister.
3. A brothers' feud in which one brother stole the other's wife.
4. Detailed lists of where some people lived and for how long.
5. Descriptions of events in a man's military service (marched from X to Y, etc).
6. Depositions by family members who were previously unknown or that narrowed their time of death.
7. Confirmation that one of my relatives was indeed a bigamist. The pension examiner did a long investigation to ferret out the whole story.
8. A description of the food produced one season on my ancestor's fruit farm.
9. A deposition providing the only known piece of evidence that three brothers briefly lived together after the Civil War.
10. Evidence of a previous marriage and death of the first wife when no other records were found.
While viewing the records in person is an amazing experience because you are actually touching the same pieces of paper as your ancestor, a trip to Washington, D.C. is not feasible for everyone. Fortunately you can order pension files online through NARA's website here after filling out form NATF 85 (for federal service through 1903). If you already know the unit your man served with you can search the Civil War and Later Pension Index available here at FamilySearch.org. Unfortunately, the actual images from the Family Search index are hosted by Fold3.com so the only free option is to access the records at a library with Ancestry Library Edition.
There are three reasons for finding the pension card: 1) so you know if your man actually applied for a pension, 2) to better identify your soldier when there are several men with the same name (such as by finding a widow's name on the card), 3) to obtain the pension application number for inclusion on the form. You don't actually need to include all of the requested information, but the more you do the better chance you have of obtaining the specific file you want, especially if your soldier had a common name. I encourage you to download form NATF 85 to can see what information you should try to locate.
Once you are ready to order you have a couple of options. You can choose the complete file (up to 100 pages) for $80 (for Civil War and later, $55 for pre-Civil War files) or for $30 you can receive a packet of eight genealogically relevant documents (death, marriage and birth certificates, if present). As tempting as it is to try to save some money and go for the documents packet, I would highly recommend selecting the full file because otherwise you risk missing the details found in the depositions. For more information about the process, download form NATF 85 here which has additional information.
I don't remember how long it took to receive my copies, but I believe it was 3-6 weeks, 3-6 long weeks when I was anticipating wonderful findings. However, if one of the files you requested wasn't found and you know for certain from the pension index card that your man filed, don't despair immediately. Wait a year or two and submit another request because the record could simply be misfiled. I once received a file on my third request. Some files, sadly, are MIA. The file I most wish to peruse is unaccounted for. Philo Brown is the only one of my soldiers who was wounded in battle, when a minie ball tore through his forearm. The only reason I know this is because of information I found in his compiled military service record. I may try again, but after three strike-outs I'm losing hope it will ever turn up.
Overall, I would have to say that I've had good luck. Of fifteen pensions I have requested, I have received fourteen, all but one on the first attempt. In all but two cases the files included information through the death of the claimant. I'm glad I discovered these wonderful resources early in my genealogical adventure. Once you look through your first one, you may find it difficult to stop at just one.
Note that the Civil War pension records held by NARA are for men who served in the U.S. military for the Union. For records on men who served in the Confederacy you will need to contact the appropriate state archive. Contact information is available on NARA form NATF 85.