I absolutely love dips. There are a couple things that could be going on. Dips are hard, and the fact that they are hard can lead to form issues, the same way that trying to squat heavy or pull too much can lead to form issues if you aren't ready yet.
Here's how most people do, or atleast attenpt to do, dips;
Notice that the torso is upright and the shoulders are shrugged up to the neck. It even looks painful and is obviously not an advantagious position from which to produce maximum force.
Here's another example-
Unfortuantely, these examples are from bb.com and wikipedia, respectively. This is unfortunate because these are the resources people are using to teach themselves how to do these exercises.
Now lets see a better way-
Notice the differences. Instead of the shoulders shrugging up to the neck, the shoulders are pulled back and the chest is stuck out forward, just like a good bench press. The main difference is that the chest and head are now looking DOWN instead of forwards. In this case this happens because the chains around the neck move the center of gravity towards the head, but chains are not required to get this better form.
Here's another example-
Although this crossfitter is actually performing a muscle up, which is a pullup the goes directly into a dip, you can see that her knees are tucked up underneath her. This also shifts the center of gravity and allows you to face downwards. You can also see in the example just how low you can get with better form, which puts your shoulders in a much more natural position in the stretch portion of the movement.
This change, tucking the knees up towards the chest, also turns the dip into a chest and tricep movement, instead of just a tricep movement.
One interesting thing about bodyweight training to remember is that we can't just lower the weight. Remember when I said that doing dips can be like squats that are too heavy before you're ready? Well, you can lower the weight of the squats and work your way back up with better form, but you can't do that with bodyweight stuff. The trick is to use a good spotter, which in the case of dips or pullups means getting a spotter who does nothing at all. He should stand behind you and hold your feet and nothing more. Then it's up to YOU to determine how much help you give yourself with your legs.
Good god, the Army got something right-
Notice that the spotter isn't doing anything. It's up to the person doing the exercise to determine how much of their legs they want to use to help themself. At first reading it might seem that someone will just use their legs to cheat the whole time, but you'll be very surprised at how well this actually works.