The word "tangent" comes from geometry (please google images of "tangent to circle"). Basically, when a line touches the edge of a circle at exactly one point (point A), and forms a 90 degree angle with another line leading from point A to the center of the circle, we can say that it is "tangent to the circle".
The chain on your bike comes off the sprocket "on a tangent".
If you tie a rock to a string and whirl it around in a circle until the string breaks, the rock will fly off in a direction which is tangent to the circle.
If the Sun suddenly disappeared, the Earth would fly off in a direction which is tangent to it's original orbit.
So there are 3 examples of tangents (but google images will make it much easier to understand).
Sometimes writers will use the word to describe something which goes off in a completely different direction then what was expected.
"I had a meeting with Fred but he kept going off on tangents".... means Fred didn't stick to the topic of the meeting and he started "flying off in a different direction".
"A tangent of possibility"... means that if you move off in a different direction, from the one you are expected to follow, then you will discover many new possibilities.