In order to have a chance at getting a good tackle, you must first get your feet moving in the same direction that the other person is running. It works best when you are in stride with them. This way, you can use their momentum to bring them down and not try to force them the opposite direction of their motion.
When placing your tackle, you must remember to "get low." This means, bend your knees down and aim your upper body at their thighs. This is where their center of gravity is, and it makes it easy to get them to fall.
The girl pictured below is getting low and ready to hit the ball carrier.
When you tackle, it is important to wrap your arms around the person you are tackling, that way you aren't just hitting them and letting them go by, you are attaching yourself to them. When you wrap your arms around their thighs, let yourself slide down their legs and then squeeze your arms together when you are closer to their knees. This will obviously cause then to lose their balance and stride and fall to the ground.
The image below shows a good tackle. The red player has gotten low, hit the yellow player at the hips, and wrapped around her hips.
Tackling is not done recklessly, it is important that it is done correctly to avoid injury. Most important of all is to keep your head above their hips when you are falling. Do NOT let your head end up between them and the ground. This does not feel good at all, as you can imagine. When you are the one getting tackled, its helps to know how to fall. If you just let yourself fall to the ground you are more likely to feel it, whereas, if you fall carefully "ankles, knees, hips, shoulders," it doesn't really hurt at all. It really resembles rolling into the ground.
This tackler has done the correct thing and tackled the ball carrier so that her head is on top of the ball carrier's hips.
When you are tackled, you must let go of the ball immediately. So it is good to try and place the ball so that your own team can pick it up from you. Sometimes, you may even go to ground on purpose when you know that your team members are behind you to post up on the ball. If you go into a tackle without your team member to pick up the ball immediately, or if you go to ground in traffic, a ruck will form over you.