Symmetry is when the beatmap is symmetrical in respect to an axis. The most common type of symmetry is horizontal symmetry. There are other types of symmetry too, however, like vertical symmetry, diagonal symmetry, and symmetry with respect to the origin (Normally we would say, "Think of an odd function", but that would probably alienate a large portion of the playerbase).
To guide you through the process of understanding symmetry, we are going to go through examples of symmetry in a map that is already ranked so you understand how it works. Trying to explain symmetry with just text would be futile and overall not very effective. Here is the example map we will be using (we'll be looking at the Expert difficulty; not a stellar map by any means, but it will do): http://osu.ppy.sh/s/34544
This map starts off with an early example of horizontal symmetry at 00:03:655 (1,2,3,4,5,6,7,8). This was done by first figuring out the combo we wanted to map to. In this case it was a simple combo of 8 beats. This meant that there were two easy ways to create symmetry here:
1. Have four beats represented on each side of the y-axis. 2. Have beats 1 and 5 on the y-axis, with 2,3, and 4 on one side and 6, 7, and 8 on the other side.
Here, #2 was clearly chosen. To make sure the symmetry was effective, hit circles (2,3,4) were copied and pasted where we wanted (6,7,8) to be. Then, (6,7,8) had a Reverse Selection applied to them in order to keep the spacing correct. Some newer mappers make the mistake of just blindly copying and pasting, without modifying things as needed to keep the map's flow up. This is a bad habit and is not recommended.
Another, much more complex example of horizontal symmetry is at 00:29:455 (1,2,3,4,5,6,7,8,9). First, (1,2,3) is a simple smiley face pattern; 1 and 2 were just placed an equal distance apart from the y-axis, with a symmetrical slider placed beneath them. It then gets a bit more complex, though.
The horziontal symmetry is maintained in the rest of the combo; just see for yourself. These types of patterns typically involve a lot of experimenting to get just right. Here you'll notice that the spacing was almost perfectly maintained; a jump was not wanted here, so keeping the spacing correct as a necessity. Doing this can take a while, but is often well worth it. Also, you'll notice that (6) is technically not symmetrical. However, having it there makes the pattern funner and flow better without destroying the pattern, so in it goes.
You'll notice more examples of horizontal symmetry in this and other maps. Just experiment and see what works for you.
The very first combo in this map is an example of lazy vertical symmetry. A line like this is the simplest symmetrical pattern you can make. It's effective when used correctly, but only use it when it works.
00:54:055 (1,1) is a traditional example of creating vertical symmetry by utilizing sliders. As you can see, it's both simple and effective. Again, don't overuse it, but it's a simple pattern to use when you are having trouble thinking of anything better.
Diamonds like 02:04:705 (1,2,3,4,5) are symmetrical both vertically and horizontally. It's a bit lazy, though, which is why (6) is placed where it is in order to twist the pattern a bit. Twisting up traditional patterns make you look less lazy than you actually are, and are typically a great way to start creating patterns of your own.
Symmetry With Respect to the Origin
-f(x) = f(-x)
This kind of symmetry is sometimes called "rotational symmetry", but that would be incorrect. It's simply symmetry with respect to the origin. It's simpler than it sounds, though. This type of symmetry is done by first copying and pasting a pattern, and then flipping it both vertically and horizontally. This type of symmetry is nice because it's more subtle than horizontal or vertical symmetry.
00:06:055 (1,2,3,1,2,3) is one example of symmetry with respect to the origin. It fits with the music; you'll notice how it, in a way, goes with the changing pitches. Due to the changing pitches, horizontal or vertical symmetry might not have been anywhere near as effective in this case as symmetry with respect to the origin was.
02:20:755 (2,3,4,5,6,7,8) is a slightly more complex example. You'll notice that (2,3,4) and (6,7,8) are symmetrical. What about (5), though? (5) is also symmetrical with respect to the origin, even though there was no copy and paste involved with that one. Go ask your math professor for more details.
Modified Symmetry is when you take a symmetrical pattern and modify it in various ways. Sometimes it works and sometimes it doesn't.
00:20:455 (1,2,3) is one of the more successful examples of modified symmetry. It uses the Scale By feature in order to make (3) a shorter version of (1). It's successful because it's actually noticeable. Some people like this, some people don't, though, so it's up to you.
01:08:455 (3,4,5,6,7) is a failed attempt at modified symmetry. Here, the left slider is part of the right slider, with (6) attempting to "complete" the left slider. To put it simply, it failed because few people noticed it; many modders thought that the mapper was just freestyle it here.
This map has other good examples to use; just look through it. However, if you want to look at other maps, Krisom's maps are considered by many to be a stellar example of good patterns and structure, with some good examples of symmetry involved.