The backbone of rhythm is a periodic beat. Usually, musicians count four beats at a time: one, two, three, four, one, two, three, four. This is called 4/4 time, because we are dividing the music into four equal quarters. As such, each individual beat is called a quarter note, and we call each instance of four quarter notes "one measure".
As you might imagine, we can divide one measure into different subgroupings. Here is the same measure divided into 8—these are called eighth notes. Here it is divided into 2—these are called half notes. Here it is divided into 1—this is called a whole note.
Whole notes, half notes, quarter notes, and eighth notes are all powers of two. We can continue this pattern as much as we'd like: into sixteenth notes, thirty-second notes, sixty-fourth notes, one-hundred-twenty-eigth notes, and so on. Of course, practically speaking, thirty-second notes are the shortest duration you will generally see, and even then only rarely.
Let's go back to our one measure of 4/4. We can also divide this measure into a grouping of 6. We call these "quarter note triplets", because we're now playing three quarter notes in the span of two. Similarly, here are eighth note triplets, sixteenth note triplets, and half note triplets. While triplets are used sparingly in Western classical music, they are used much more frequently in many popular styles of music, like jazz and hip-hop.
Here is a different way of dividing a measure of 4/4. As you can hear, the lengths of the notes are no longer equal. The first note has the duration of a quarter note and an eighth note combined. We call this a "dotted quarter note", because in written musical notation, the quarter note is followed by a dot. The second note is also a dotted quarter note, and the third note is a regular quarter note.
Here is another way to visualize what is going on. The top line shows our measure of 4/4; the bottom line shows the current rhythm. Pay attention to the fact that the bottom rhythm's second note does not align with any quarter note in the top rhythm. Therefore, we say the bottom rhythm is "syncopated". The first and third notes are on-the-beat, while the second note is off-the-beat.
In addition to dotted quarter notes, we can also have dotted eighth notes, dotted sixteenth notes, and so on.
Another important element of rhythm is silence. Here is a more complicated rhythm. This rhythm is two measures long. The first measure consists of two dotted quarter notes followed by a regular quarter note. The second measure starts with a quarter rest, then two quarter notes, and then another quarter rest. Rests may have the same duration as notes, but instead of there being sound, there is silence.