Aloe vera is a very common houseplant in colder climates and often grown as a garden perennial in warmer ones. It has also escaped from cultivation and become a wild plant in many places (it's natural habitat is unknown, probably somewhere in Arabia, as its been cultivated since time immemorium). Not only is it attractive and easy to grow but it also has medicinal properties and a very short and easily remembered Latin name.
"Aloe" is the genus a group that normally includes several types of similar organisms and it should have a capital letter. The genus Aloe has about 400-500 species and numerous hybrids. "vera" is the species a group of very similar but probably not identical organisms (e.g. all humans are the same species: Homo sapiens) and should start with a lower case letter. The Latin name should be underlined or italicized. "Aloe vera" is NOT the generic term for any plant with rosettes of fleshy leaves! There are lots of leaf succulent rosette plants in a number of different families and even a few cacti that have large leaf-like tubercles (bumps on the stem) to really confuse you.
The first step is to make sure your plant actually is an Aloe vera as there are many plants that look rather like it. It's not that the occasional look-alike accidentally creeps in among the Aloe vera plants. Many of these leaf succulent rosette plants are perfectly good houseplants or garden perennials for warm climates so garden centres sell them (and seed companies sell the seeds) and people (even ones who know what these plants really are) grow or even collect them. It may well be more the case of picking the Aloe vera out from the other Aloes, Gasterias, Agaves etc.
Aloe has a (or often more than one) rosette of, pale green fleshy leaves with a few white spots on a short stem that's normally hidden in the middle. The leaves curl upwards to form an urn shape. It grows up to about 1m tall but is normally 20-30cm. There are small whitish teeth round the edge of the leaves. It is not a cactus and therefore does NOT have hairy patches (areoles) thus eliminating cacti with leaf like stems (e.g. Epiphyllum and Opuntia) or leaf like tubercles (e.g. Ariocarpus and Leuchtenbergia). The leaves on small plants are in a fan shaped arrangement (discidous) but if your plant has leaves 10cm long or more, still in this arrangement, you've probably got a different Aloe or a Gasteria. If you look in the centre of the rosette, you should be able to see progressively smaller leaves spiralling down to nothing. If the central leaves are rolled together to form a cone, you've almost certainly got an Agave. Other Aloes vary hugely from small grass like plants to sizable trees but they often have darker leaves and/or more vicious teeth than A. vera and grow somewhat larger. Aloe vera has yellow, tubular flowers that don't open up much at the ends. Almost all Aloes have flowers that are a similar shape. It isn't the only yellow flowered Aloe but other species have various flower colours. An Aloe like plant with flowers that open out flat is probably a Bulbine.
There are two main version of Aloe vera (apparently there are about 150 clones although each time one grows from seed it should create a new clone but not necessarily one that will be vegetively propagated and circulated). The true A. vera is larger with darker leaves and you normally buy it from a garden centre. A. vera var. chinensis (a plant of mysterious origin) is smaller with paler leaves and seldom seems to flower. A. vera var. chinensis also clusters more prolifically.
Aloe vera is a very easy plant to grow. It's a desert succulent and standard succulent treatment will probably suffice. If you already have other Aloes, Gasterias or Haworthias you can probably grow it the same way. In fact it will probably grow well if you treat it the same way as your desert cacti (treating an Aloe like a jungle cactus probably won't work well) or Crassulas. There are a few differences from normal succulent treatment. Aloe vera (like most Aloes) will tolerate more shade than most succulents so you could grow it in a north-facing window in the northern hemisphere. It needs to be protected from frost. Most cacti and some other succulents will survive frosts that will kill Aloe vera and most other Aloes. It doesn't like perlite. I have a number of Aloes and Gasterias growing in a mixture of perlite, gypsum and sand but Aloe vera doesn't like this. A mixture of standard potting compost and sand is probably best. They should be watered (a good soaking) once a week in warm weather (or if kept in a warm place inside) but can survive a long period of drought. They'll probably grow better if you neglect them than if you fuss constantly. The pot should have drainage holes. Like all plants they need some nitrogen, potassium, magnesium etc. but don't need much. I know somebody who fertilized hers and had them growing like crazy.
Most pests don't like them but I've found that cats can nibble the leaves (particularly of var. chinensis). White scale is the scourge of Aloes and Haworthias (but they don't seem to like Gasterias). Chloorpirifos seems to be the recommended treatment for this. Over watering can cause rot. You should cut off the rotted parts. If this leaves you without roots, you can re-root what's left assuming there's some stem.
Aloe vera is normally grown from cuttings. The offsets can be yanked out, left to dry for a bit and planted. Like most Aloes, it doesn't grow from leaf cuttings. I've never tried growing it from seed but most Aloes grow from seed by normal succulent seed raising methods. Being a monocot, it can't be grafted.
So just go and neglect your Aloe vera plants and they'll probably grow OK.