Twin flames are any pair of human beings whose souls were once conjoined at the time of their creation by the gods and subsequently separated by the same. They are essentially one soul in two bodies and thus often complement each other, having qualities the other lacks, such that their reunion brings about great peace, love, and joy. Twin flames are often at their best when together and can accomplish much more than either of them can on their own.
Twin flames are not the same as soul mates.
Soul mates are any two souls who - whether by accident or design - happen to meet over the course of multiple incarnations. That is, they cross paths across more than one lifetime. They can be acquaintances, friends, lovers, parent and child, enemies, or other relations.
All twin flames are soul mates, but not all soul mates are twin souls, since only twin flames are pair-bonded one-to-one, whereas a person can have any number of soul mates over the course of their existence.
Twin flames often, though not always, reincarnate in tandem with one another. Often, it is desirable for them to do so in order to maximize their chances of finding - and, more importantly, recognizing - one another. As their name suggests, they are twins in so far as their souls are the same age across incarnations; though, depending on the circumstances of their lives within any particular incarnation, they might die at different times and thus have to wait for a time in the afterlife while their partner catches up, or else run the risk of there being an age gap between them when they find them again on the material plane.
One way in which twin flames can recognize each other is they will have matching Thelemas. However, if one or the other is not born with a Thelema, or otherwise converts to a religion that doesn't have them - such as the Ring of Saturn - then this approach becomes rather futile.
Such complications can often confound twin flames in their quest to find each other and are part of the reason - along with geographic and temporal distance, as well as life circumstance - for why they might ultimately fail at this within a particular lifetime.
Fortunately, as twin flames grow wiser, they can capitalize on reincarnation to improve future searches.