tl:dr royal people used to get in on out of wedlock quite a bit. How likely could someone be a descendant of an illegimate child originating from them?
Let's just stick with European royalty since most people tend to be at least somewhat more familiar with them.
My mom claims that my she is descended for an English royal bastard child. She says the child was smuggled over seas to Canada for safety reasons. She even claims that money used to be sent to my family from over seas from an anonymous source.
I'm pretty skeptical of this since the majority of my mom's family came over during the Irish potato famine and she only has a small number of English ancestors.
However it's not like illegitimate children didn't exists, many nobles had extra-marital affairs. How common were these children and how likely is it that their descendants would still be kicking around today?
I'm not sure I understand your question... Are you asking about the plausibility of your mother's story? Or are you just wondering whether or not royal bastards went on to found their own lineages?
If it's the former, I'm afraid that it sounds an awful lot like fantasy -- while royal bastards have always been a fact of life, especially when you consider that royal marriages were historically politically motivated, and almost never took into account whether or not the two parties were physically or romantically attracted to one another, the whole "smuggled to Canada for safety reasons" sounds a little too Hollywood if you ask me. Maybe it's true, but without any more documentation like the identity of the royal parent, chances are good that it's just fiction.
As for the second question, absolutely. Royal bastards were frequently used by their royal parents in marriage alliances with foreign powers, so being illegitimate was actually not a huge problem for most of them throughout the scope of European history. You only have to look as far back as Charles II to see that he recognized 14 of his 20-odd illegitimate children, and most of them went on to establish some of the most prestigious noble families that are still among the upper echelons of the British elite. Princess Diana was descended through two of Charles' bastards, for instance. Wikipedia has a nice little chart of the noble families that can trace a direct path back to one of Charles' bastards if you're curious.
Since we're talking English/British royalty, the fact that the vast majority of kings had at least one recognized bastard meant that on the whole, all that mattered was that the king claimed the child as his. Didn't matter if the kid's mother used to sell oranges in the theatre pit at Drury Lane (like Nell Gwynn, whose son by Charles went on to found the Dukedom of Saint Albans, which is still in existence to this day) or the wife of a nobleman like Barbara Villiers, Charles' primary mistress.
There are only a handful of problems for a royal bastard that could occur that would cause any major drawbacks to being the illegitimate offspring of a king. The primary issue to be avoided was developing any designs on the throne -- Charles II's first son, the Duke of Monmouth, got it into his head as a young man that he should be named Charles' heir, despite his mother being Lucy Walter, a commoner (there was a faction that supported the idea that Charles had married Lucy while he was in exile, which would make Monmouth legitimate and therefore legally Charles' heir). The fact that no marriage could be proven absolutely excluded him from the succession and Charles spent a great deal of effort trying to convince his son to give up his ambitions but Monmouth persisted and eventually started a rebellion that lead to his execution in 1685. Given that Charles died without a legitimate heir to the throne (his marriage to Catherine of Braganza was childless), you'd think that if there was a chance to prove his son's legitimacy he would have taken it... But the fact that a marriage could never be proven to have taken place and that Charles denied it vehemently speaks for itself.
The other issue, which to my knowledge hasn't ever definitely been proven, is if the child was the offspring of a reigning queen by another man other than her husband, the king. There's a fringe theory that Elizabeth I gave birth to a child in her teens by any number of possible suitors at the time (including Robert Dudley and her stepfather Thomas Seymour) and the child was raised by another family under an alias. This theory is not taken seriously at all by the vast majority of historians, but it does point to the problem that arises when a female monarch has a child out of wedlock -- it puts the succession in dispute and would allow for the theoretical father's family's claim to the throne to be taken seriously. While a commoner woman giving birth to a king's bastard presents very little risk to the established succession because women < men at this point in history, a commoner man could in theory cause a whole lot of drama if he sired a child with a reigning queen.