This is a question that will not lend itself to a simple answer, because the scope of it is larger than it may first appear.

The Jacobite century is often counted from the Glorious Revolution in 1688 to the death of Henry Benedict Stuart in 1788. It comprises multiple conflicts, including the Williamite Wars, Dundee's rising, the First and Second Jacobite risings, a small one in 1719, multiple riots and a long period of unrest, all of which affected Ireland, Scotland and England differently. In each of these countries, as well, support waxed and waned for a myriad of reasons. I focus on Scotland (and much more on the '45 than other periods), so I will leave aside much of the detail on Ireland and England.

As you might expect, support runs along political lines and grew at times when groups felt they were being treated egregiously (I've written in more detail on this here, or when Stuart's core supporters felt they had the upper hand.

The Jacobite court and Jacobite operatives were frequently in negotiations with half of Europe, trying to convince them to come to Scotland's aid and restore the Stuart crown. Mostly this fell on deaf ears, but the '15 and '45 risings did gain foreign support as part of the greater Wars of the Spanish and Austrian Succession, respectively. Fun fact, part of the Treaty of Utrecht that ended the War of the Spanish Succession specifically stipulated that France HAD to kick the Old Pretender and his court out of the country. The very abortive 1719 rising was supported by Spain as part of Philip's attempt to reassert Spanish power following the War of the Spanish Succession (by destabilizing Britain). Both the '15 and '45 had planned French support, though this fell through in each case.

After the '45, brutal tactics were employed by England to finally quash any support for the House of Stuart, but periodic riots in the name of the ousted king still cropped up in the decades that followed. When Henry Benedict Stuart, Charles Edward's younger brother, died (after having served in the Catholic church for many years), this laid to rest most support for the Stuart cause, as Charles had but one bastard (late in life legitimized) daughter. However, some do still recognize a Jacobite succession today.

I hope that gives you a bit of insight into your question--a book would do far better justice to it than I could, and Daniel Szechi's The Jacobites is a good primer text that doesn't assume any prior knowledge. (Caveat: I do find he is rather unsympathetic to his subjects in places, but it's still an excellent book.)