A concept very closely related to invariance is symmetry. In fact, symmetries are what currently guides us most in the construction of theories of elementary particles.
A symmetry is in the beginning the fact that something looks similar when viewed from different perspectives. Take a ball, like a snooker ball, but paint it only in a single color with no markers. Then, no matter from which direction you look at the ball, it always looks the same. Or, you can turn it as you like, it always looks the same. The ball is just the same from all directions, a perfect sphere. Thus, it is called to be symmetric under a rotation. Therefore, this symmetry is called rotational symmetry. With this already the link to invariance comes in: The ball looks the same from all direction, it is invariant under the position of the one looking at it. There is always an invariance when there is a symmetry.
If you start looking around, you will find symmetries to be a rather general concept. If you take a blank sheet of paper, its front and back look the same: It is symmetric under flipping it from front to back. Or take a snow-flake. When looking closely, it has a structure with six rays. Thus, if you rotate it by a sixth of it circumference, it looks like without rotating. Both these examples are so-called discrete symmetries. For the ball, we could rotate it arbitrarily little, and it still looks the same. Not so the snow flake. If we would rotate, say, by a tenth of its circumference, it would be obvious that someone rotated it. It only looks the same when rotating it by a sixth of its circumference. There is only a finite number of things we can do to it to make it look the same, while there is an infinite number of things we can do to the ball.
To find another example of a symmetry like the rotational symmetry, which is also called a continuous symmetry in contrast to the discrete symmetry of the snow flake, imagine empty space. If there are no stars or galaxies or so, then you could move a step to the left, right, front, or whatever, or half a step, and whatever you do, it always looks the same. This is the so-called translational symmetry. Moving you in another direction just gives the same result. You could also rotate yourself in space, without changing anything. Thus, you can combine the rotations and the translations to a bigger symmetry, a so-called product symmetry.
What is, if there are two people in outer space? Now you cannot move alone, and everything is the same again, because the other did not move. However, if both of you take a step of the same length in the same direction, nothing appears to be changed. In this case, one says that the symmetry is only applying to the complete system: When always moved together, the two of you form a system, which is symmetric under common translations and rotations.
Another important concept with symmetries is that of an approximate symmetry. Take a person. The left-hand side and the right-hand side of her face look at first symmetric. You could just mirror them, and it would look the same. This appears to be a discrete symmetry, actually a mirror symmetry. However, if you look closely than the person might have a slightly different shade of eye color on the left than on the right. Thus, though it looks almost as if there is a symmetry, it is actually not there, but almost. This is an approximate symmetry. If, for example, the person would have painted her face on one side blue, then the symmetry is not even approximately there, it is just different. In this case, one also calls it a broken symmetry, broken by some external effect, here the painting. Symmetries which are not flawed in either of these ways are called exact. The snooker ball had an exact rotational symmetry. Would we have left the number on it, the symmetry would have been broken.
This is already a long number of different types of symmetries. There have been continuous and discrete symmetries, the symmetry of a system and the individual symmetry, product symmetry, an exact, approximate, and broken symmetry. If you go around, you will easily spot more of them. A sausage shows a symmetry when rotating it about its length, a leaf of a tree has a mirror symmetry like a face, and so on.
In elementary particle physics, it turns out that symmetries are deeply connected to the properties of particles. For example, each force can be connected to a symmetry. The fact that we have mass can be traced back to a broken symmetry, as that there is more matter than anti-matter. And this is just a short excerpt. However, to really understand these, it requires another concept, the difference between local and global.