Two Ply yarns are very different from singles even though they are made up of two singles. Essentially they are created by aligning two singles and allowing them to release their twist against one another. They are often referred to as being in balance with one another. This balance avoids the bias problem which can occur with singles. Plying controls the twist and evens out the visible inconsistencies in strand thickness, since the differences are averaged out over the finished yarn.
Most singles are spun with Z-twist, by
turning the fibre clockwise. When plying, the yarn strands are turned counterclockwise, so the finished yarn has what's called an S-twist. This can also be done in reverse.
The amount of twist at each stage can be very different depending on the source fibre and on the result the yarn creator is looking for. No matter what variations are used, the yarn becomes noticeably stronger. For that reason, smoother, weaker than wool fibres benefit from being plied to add strength.
The twisting of the plies together mean that the strand formation is now more oblong than round. (I searched for an image that would demonstrate this when viewed as a cross section and didn't find anything on line.) When knit, the two ply yarn produces a textured surface, especially when compared with singles. If you don't have any examples in your own knitting, try taking a look at Ravelry photos of projects worked with singles and compare them with projects worked in 2-ply. I don't want to pull any photos for my blog from Ravelry without anyone's permission. My single ply work is textured but you can see one here:
I don't have an example of a 2-ply project any more but it's generally considered to be an excellent yarn for lace work, as the shape and opposing twist of the strand holds the fabric open. In heavier weights it has a textured, pebbly, more rustic appearance. You will see what appears to be slightly uneven gauge even with yarns which have a halo, but it is due to the nature of the yarn not the knitter.
It's difficult for the average knitter to just look at a plied yarn and know what version it is. Take the cut end and untwist to separate the plies.
If you have any spinners in your knitting circle they can be great sources to aid you in understanding the spinning process and can help you to expand your yarn vocabulary.
Part 1 is here. Part 3 is here.
If you want more understanding of how spinning and yarn structures work, check out this Craftsy blog post.