Heirloom can mean a lot of things. Typically heirloom vegetables will be open pollinated varieties that have been around awhile. How long that awhile is will vary. Heirloom will also be used synonymously with open pollinated(OP), which it isn't.
If you desire to save your own seed then you need to raise OP varieties.
Are they better? That depends on what you mean by better. If you are a large commercial farmer that grows hundreds of acres of tomatoes and needs to pick them all at once and ship them hundreds or thousands of miles then the characteristics of riping all at once, all the same size, and having built in packing materials - solid and tough - then an heirloom most likely isn't for you.
Many times heirlooms were bred by either a university for the gardener or was a result of a family saving seed from one year to the next. If someone was saving seed from year to year they where going to save seed from plants that they liked to eat and that did well in their area. The farthest they were ever going to be shipped was maybe 10 miles to the local farmers market in the back of a pickup or wagon. They are generally thought to taste better. Many times they will ripen over a longer period that a newer commercial variety.
It doesn't have to be an either one or they other choice. Devote some of your space to heirlooms and keep experimenting. If one heirloom doesn't work for you try another. There are literally hundreds and thousands of varieties or heirlooms.
Here is a link to a similar discussion going on, on the veggie forum. Apparently, things are not as they appear. If your not a subscriber you may not be able to access this link. But I think most of you will, at least at the moment. Edited because I posted the wrong link,http://davesgarden.com/community/forums/t/1251285/. Hopefully this is correct.
As for "heirlooms", 50 years old (or older) was/is the usual given time/age to be considered heirloom. I haven't heard that "anything before 1971" was considered to be such; that's very young in the heritage/heirloom world.
"I keep seeing this term used in many forums, but I'm not sure what it means. Can someone please explain it? Are they better than 'regular' vegetables? Should I be using them instead"
As for "better", please keep in mind very few things as a group are ever considered better. In this case some heirlooms might be better than some hybrids and some hybrids might be better than some heirlooms, but as a group not all heirlooms will be better than hybrids nor vice-versa. Keeping "better" in mind, there is taste to consider (and all of us taste things differently so that comes into play); there is growth rate, production, color, disease tolerance (if any), etc that all come into play.
Best to just jump in and try some of each cue-chick. What a journey you're about to embark on.
An heirloom is usually considered to be a stable variety. It has been grown for many years with the same results. A hybrid variety (which are very common now) is a cross between at least two different plants of the same cultivar. Sometimes more than one generation of crossing is involved, think grandparents and great grandparents. If you collect seeds from the F1 generation/plant (what you are buying as seed from the manufacturer) of a variety the results that you would get in the F2 generation (the second year, with seeds collected from the first year) will be highly variable. You may get offspring that are nearly identical to the parent or plants that are radically different. The more cultivars of the same plants you have in the yard the more likely you are to get variation because you simply do not know which plant your vegetable was pollinated with.
Hybrids are very common because manufacturers are trying to artifically select for specific characteristics: color, flavor, resistance to disease, ripening time, number of seeds, etc. etc. etc. Think of it as nature in fast forward. What may have happened over hundreds of years (or never due to geographical distance) may happen in a matter of seasons.
That being said, heirlooms are delicious! When too much science is involved flavor and variation is frequently lost. Store tomatoes all taste the same. There's a heirloom for every tastebud out there. I can barely try them all fast enough!