To begin with I want to talk about the 125cc 2-cycle dirt bike. Now there are people out there who love 2-cycle and won't ride a 4-cycle bike, but that doesn't matter now. I am just trying to relay a few facts. The 125 is usually set up to ride like a bigger bike, with a higher seat, and bigger tires. Many people think these are great bikes for up and coming riders who may be young and can benefit from the higher seat, but not too high and the excellent amount of power, but not to powerful. This also works great for older riders because the bike is big enough for full grown riders, but is not so powerful that it may be a chore to ride.
Next is the 250 4-stroke. This is much like the 125, but a smoother ride. 2-cycle bikes run at a much higher rev than 4-strokes which cause them to be more powerful at less cc's. Thus the 4-cycle 250 is going to perform like a 125 much have a longer power band. So this would be good for racing or riding fast for a long amount of time whereas the 125 would have been better for trail riding or a lot of top end power.
Now the 250 2-stroke is a favorite of many riders out there, and especially the older riders. This bike has great top end power and also a good deal of lower end power. Great for racing and trail riding, it is still the preferred bike for freestyle likers due to its lightweight make-up. This bike is for the advanced rider due to the touchy throttle power and it generally high seat height and big tires making it a little more difficult for shorter riders to ride when the bike is stock.
Finally the 450 4-cycle. The 450 is a newer idea that just came around in the last decade or two and is gradually taking all the glory these days. All professional supercross and motorcross riders are riding 450's due to their immense power at top end and low end. Unlike the 250 2-cycle the 450 is able to run it gears way longer and is thus a lot less work to ride while racing due to the minimal amount of shifting required. The only thing I have to say about the 450 is that it is awfully have and thus more awkward to handle if you like riding trails in the woods, and thus for that I would suggest a 125/250 2-stroke since they are lighter than their counter parts.