Strategy is basically just a plan. When you're racing, you can have a very basic strategy, such as your plan to run at a certain pace or exertion level, or your fueling plan.
One of the good things about having a competetive strategy is that it helps you to keep your head in the race. If you've done much racing at all, you've almost certainly had times where things start to get really tough, and all you want is for the race to be over. At that point it's easy to let your mental edge slip away, but sometimes there are things that you can do to help with that, and working on your race strategy can help.
Trail racing strategy can be quite different from road racing. When you're racing on the road, for the most part you lock into your pace and try to hold steady as the miles click by. If you're more competetive, you may be making moves on your competitors as well, but the pace tends to be a lot more even when on the roads.
When racing on trails, your pace is going to be all over the place, so having a target average pace in mind can sometimes be not be that useful- you'll be fast on the downs and flats, and slower on the ups. One of the key things is to know what your strengths are and play to those. In my case, I'm not that great on climbs, but I feel much more comfortable going fast downhill, so I tend to save my energy on the uphills and then really push hard when going down. It's important to know what your strategy is for that- it can be tempting on an uphill to try to keep up with those around me, but I make a concious decision to do my own thing knowing that I'll have a decent chance of making up that time when we start descending.
Something I do to keep myself mentally sharp when trail racing is pay attention to the runners around me- I've learned some pretty good strategy from observing.
One thing that I've found really effective is using hills strategically. If I'm coming up on someone that I feel like I can pass, and we are climbing, I tend to stay just behind them. When you have someone right on your heels, you tend to speed up as it feels uncomfortable to have them right there. If they do speed up, and you stay at a comfortable pace, they'll feel worn out at the top of the hill, and you can then pass and open up some space. I do the same if someone comes up on me on a hill- I just hold my pace and don't let myself get tricked into going faster, so that I can open up at the top with the hope that they won't be able to match.
Another thing I do when racing on trails is look for footprints in the mud. If you're far back in the pack this won't be too effective as there will be lots of footprints to look for, but if you're near the front it can be helpful. A few weeks ago I ran a 50K, and at one point on the course I thought I might have taken a wrong turn, so I was looking for the footprints of the leader (I was in 2nd). I ended up spotting one (I had noticed that he was wearing La Sportivas) so I felt more confident that I was going the right way. I mentioned that to him after the race and he told me that he makes a concious effort to try to avoid leaving footprints so that those behind him won't have any sense of how far ahead he is. I thought that was interesting.
Trail racing is such a different sort of thing from running in road races. I find that the concentration that it takes, the varying scenery, the varying pace at which you're moving all contribute to making it so fun.
If you haven't yet tried a trail race, give it a try this fall. They tend to be low key and welcoming, and it will give you a chance to try out something really fun and different from road racing.