It appears to me that most major DJ's already published their song and are replaying it with some effects during a live show. When a DJ such as Skrillex is on stage and is mixing, how much work are they actually doing?
It depends on the DJ and many variables like: his skill set, his knowledge, his work, his selection of songs, etc.
There are many different tools and techniques out there, and DJing can become as complex as you want. These are some of the things that a DJ might be doing while carving and performing a set:
In order to mix and blend two or more songs, those songs need to be at the same speed. If song X is currently playing, and its speed is 128 BPM (beats per minute), and you want to mix it with song Y that is at 127 BPM, you need first to make song Y faster, you need to make it playback at 128 BPM.
This is very simple to achieve if you DJ using software, like Aleton Live, Serato, or Traktor. You press one button, and the songs will sync automatically. It is harder with turntables (vinyl) or CD players, as you need to sync the songs manually. Doing this manually requires a lot of practice and skill.
Once the songs are at the same time, now they need to be in the part where you want them to be. Also, the elements need to be aligned. Two tracks that are at the same speed but not correctly aligned can sound very unpleasant. Some percussive elements like the drum bass need to sound at exactly the same time, with very small room for error.
Again, this is much more easier using software.
Planning and executing your DJ set using music theory. Which keys will you mix? In which order? You can make chords, scales, progressions.
There is a thread here about it: What is harmonic mixing?
Some DJs plan their whole sets around the keys of the songs. Some ignore harmonic mixing completely.
When and how you mix two or more songs. Which songs/elements are you mixing? At which parts?When are you adding the songs/elements? When are you killing them? Are you using filters? Will you use effects? Will you use volume? This, along with the song selection, is what really carves a DJ style.
This can be as simple as crossfading from one song to another (even without beatmatching!) or it can be much more complex. Some transitions last several minutes, while many different songs and elements are being manipulated in different ways.
Making transitions sound good is itself an art.
Which songs will you play? When? Why? Some DJs have a predetermined ordered set that won't change or will change a little during its execution. Other DJs have nothing but a collection of tracks, and the order is dictated by the people that are listening: do they look/sound like they want me to play something more intense? Do they seem to want something smoother?
Some electronic artists and DJs add live elements to their act. Guitars, drums, voice, whatever you can think of.
These are by no means the only dynamics you will find on a DJ set, but it's a good introduction to "what's going on". How much work do DJ's do? Some do nothing, some do a lot, some do a lot more.
I've seen respected DJs get away with playing a pre-recorded set and acting like they are doing something.
Some DJs have very complex DJ sets conceptually, but not mechanically; like a simple setup of 3 turntables and a simple 3 channel mixer with 3 band eqs (simple mechanically, in a sense) but with many complexity in harmonic mixing, transition, selection, etc.
Some DJs have very complex DJ sets mechanically, but conceptually not as complex; like a setup of 2 or more DJs with 4 turntables connected to Traktor, plus 2 CDjs, plus virtual tracks, plus many control surfaces, two laptops, external analog and digital effects, loop stations, (etc etc etc), but there is not much happening in transitions, harmonic mixing, etc.
Some DJ's have very complex DJ sets both mechanically and conceptually.
The whole spectrum! Depends on the DJ. From nothing, to as complex as they want or can handle.
A DJ can be doing a lot more than just playing records into each other even though JCPedroza's answer explains that is already quite involved.
He's a video of a well-regarded UK DJ (well he's from New Zealand but works here):
You'll notice his hands never stop moving. I don't really know what he's doing but for instance he will typically loop a song and sample it, building up a unique composition on the fly. Other things look sort of similar to a guitarist doing tone/volume swells - swelling the volume or the panning in time with the beat, etc.