Blow Em Up
To start, at a primal level, you can consider using that built-in pump that you were born with, your lungs. However, from many standpoints, this is mostly discouraged these days.
Balloons for twisting are very difficult to blow-up and you'll have to work up to be able to do this. Most important, if you will be blowing up balloons and giving them to others, it's not considered proper to put your mouth on a balloon and then give it away.
Whether you blow your own balloons with your mouth or use a pump is up to you. But the general trend these days is to use a mechanical pump.
The first type of pump is one with a bulb. By squeezing the bulb, you produce air to fill a balloon. This type of pump often comes in beginning balloon twisting kits. They're fine if you're only going to make a few balloons. But if you have to inflate a series of balloons, you’ll quickly find these to be inadequate.
The second type of pump is a hand pump, such as those in the picture. Much like a bicycle pump in operation, you grab the main body of the pump with one hand and push a plunger back and forth with the other. The higher-end pumps feature a "two way" action which means that the pump is filling the balloon in both directions of the handle. While these pumps do vary in quality, you can usually buy them for less than five dollars.
A cheap pump will typically break with just a little use. A higher-end pump will last a lot longer. There are different pumps for different size balloons, so if you're working with 260 balloons, you'll want a pump specifically for this size balloon.
One other consideration involves the nozzle of the balloon pump. Some pumps feature a larger nozzle that requires you to physically slide the balloon's neck onto the nozzle. This probably requires less effort to keep the balloon on the nozzle as you pump up the balloon.
On the other hand, pumps with smaller nozzles make it far easier to connect to the balloon. Personal preference will be up to you, but I prefer the smaller nozzles that I think require less effort overall.
I'm going to keep the discussion here basic. For professionals, there are elaborate inflation pumps that feature motors and other setups. Of course, if you're serious about balloon twisting, you'll probably be considering these in the future.
In balloon twisting, a good pump will serve you well. With your balloons and a pump, you're ready to twist.