It is definitely a Crassula species. These are beautiful, with heart shaped leaves. It is one of the easiest of succulents (and even plants, in general) to take care of and propagate. They also sometimes have understated, yet beautiful flowers during spring/summer.
From the image, it is hard to tell what is going on. If you can take a better close up, we may be able to tell whether it is disease or just over watering, or crowding, given your plant appears to have grown quite well right now.
You could take some pictures, and then, we may be able to tell what is going on.
From this photo, it still appears the plant is quite healthy and if you cut off the dying stem heads here and there, maybe repot it to allow more room to grow, or add very diluted plant food, you might do just fine.
You should also make sure you are not over-watering the plant. The top soil needs to become dry in between watering turns.