The three items I have mentioned work in the same way, and whilst those who play only by ear rarely notice them, those who read and/or write music need to be aware of what they mean either when they see them and want to know what they would sound like, or when they wish to write something out for somebody else with one of these in it.
I will start with triplets, which are easier because more of us will have noticed them than the others: a triplet occurs when the player plays three notes that fit in the time normally occupied either one, two or 4 notes. They look something like this:-
In this drawing bar one is straightforward 4 beats of crotchets, bar two has a triplet occupying the whole bar - three notes spread evenly over four beats - and in bar three there is a triplet occupying one beat followed by three single crotchets - one beat each - making a total of twelve beats over the three bars.
A sextuplet usually happens when six notes are played in the place of either one note, two notes or four notes, and would look something like this:
In this drawing six notes are played in the time of one crotchet and are followed by three crotchets - so the six notes have to fit into the one beat, and are followed by three single notes occupying a beat each, making one complete bar in all.
And of course quintuplets happen when five notes are played over the time that would normally be occupied by either one, two or four:
Five notes occupying the time of a crotchet followed by three notes of a crotchet each, making one full bar of 4/4 time.
But we can also have some more obscure things that do occasionally happen, like:-
3/4 time with a bar that has one triplet per beat! This is different to 9/8 time in that most of the time the piece would be in 3/4 and then odd bits have triplets in them - and of course 9/8 is not written as triplets but as 9 quavers to the bar.
conclusion
One of the difficulties of trying to cover this area is that almost certainly I will miss out the particular figure for which you are seeking an explanation.
Therefore to really get familiar with these events, it would be wise to look at quite a few pieces of written music and see how people have expressed such things. They are not particularly consistent, and of course most people think their way is the only proper way! Sometimes it is not exactly clear what they want, but one can usually (but not always) resolve the matter by working on the assumption that all bars are the same length (unfortunately even this is not always the case).
I think that although I have only given an outline here, this article will enable you to interpret virtually everything you come across of a similar nature to these examples.
Good luck.