Carpet and tile often have a height difference. On a level base, most carpet will be taller than ceramic tile in look, but when you step on it, the carpet actually sits lower than the top of the tile. This leaves a sharp edge and breaks in the tile. The threshold where the carpet and tile come together can be source of tripping or damage if not properly worked with to level the two surfaces as best as possible. In order to join the two surfaces in a pleasing way you have a few options.
Carpet shims were developed to add a gradual height to the carpet from underneath. It slopes down going away from the threshold in all directions. This keeps you from tripping or breaking the edges of the tile. The solid surface can be adjusted to start as far back as you would like and to finish at the exact edge of the threshold. Shims can be used for both doorways and full seams if you have a simple door-less transition between rooms.
Some people prefer to add a small transition strip on the edge of the tile. This covers both the edge of the tile and the edge of the carpet. The strip favours the side that is taller and offers just a little slat as the transition. The strips protect both the edge of the tile and keep the carpet from fraying. The strips are not as solid as some of the other options. Once the strips have been in place for a prolonged period of time, there might be some movement. You can choose from wood strips, metal strips and even those that mimic the look of tile.
Like a shim, this transition adds height to the carpet side, but instead of ending at the exact location of the tile, it overlaps by a small amount. The carpet is then cut longer than the edge of the raise and folded over the top to make a smooth, carpeted edge to the transition. This protects the edge of the tile as well as keeping the edge of the carpeting from fraying. It does require the placement of a strip to keep carpet properly in place. This does leave a little rise, so it can be a tripping danger. This is not a good solution for a doorway.
If your tile is particularly thick, you can grind down the edge at a transition angle. This allows for the carpet to overlap directly and be glued directly to the tile. This provides a seamless transition that keeps the sharp edge of the tile from sticking up. Typically, the thinned tile under the carpet is cushioned by the overlay. Over time, the tile under the carpet may break off or separate. This transition works best for low-traffic areas.