From my understanding, inherited titles such as Dukes, Marquees, Earls, Viscounts, Barons, (Baronets?), along with owning swathes of land (landed gentry?) and a palatial house would be the prerequisites of traditionally being the aristocracy in the British terms.
If I were the second son, or daughter, of a Duke and so not to inherit the family titles or the estate, would I be considered part of the aristocracy? If I were, would my children be part of the aristocracy? And what about their children? Was there a distinct line drawn when someone would not be considered part of the aristocracy?
There are kind of two answers to this.
One is that any son who does not inherit a title or have one granted by the king, and any daughter who doesn't marry someone with a title, isn't part of the aristocracy anymore, period. Aristocracy = the peerage, and only people with titles are part of the peerage. Even baronets are technically not the peerage, but the landed gentry (untitled people with lots of land and usually money as well) as they don't sit in the house of peers: the regularly published Burke's Peerage, Baronetage, and Knightage is called that for a reason. (Burke's Landed Gentry is another book.)
The other answer is considerably more complicated. Families don't exist as tree graphs where the edges trail off into nothing - they're interconnected social networks. Let's say you're Henry Talbot (1563-1596), third son of George Talbot, the Earl of Shrewsbury. Your eldest brother dies and your second eldest inherits your father's title (you would inherit it after him, but you die before him so the point is moot). You are ... not an aristocrat, by the standards of the first answer, as you don't have a title. But you're still the son of an earl, and your daughter marries an earl herself. Is this a dip out of the aristocracy for the family, or is the network strong enough to hold you in the aristocracy? Many members of your family have titles, your acquaintances do, you're fully a part of their lives despite your lack of a title.
I'm not sure if this complicates or simplifies things, but something else that needs to be taken into account is the fact that people who do not inherit titles from older male relations can also be given them by the crown. This was possible throughout English history, usually for military or civil service, ie serving as a member of Parliament, but it also became possible to simply buy titles under James I in the seventeenth century. As a result, younger sons of great lords had a few ways that they could end up securely titled. I have a recent answer here that discusses career options for younger children in the British aristocracy.