These are the important ones:
Frame. It could be wood or metal but sturdy enough to hold the playing area firmly so that it does not sag. In particular, slate surfaces should have solid wood cross-bar frames to support it firmly.
Top rails. Wooden rails are better than particle board because they are long-lasting, they hold the cushion material more firmly, and offer a better rebound to balls.
Rail cushions. Ensure that they are A-grade K-66 natural gum rubber bumpers, the industry standard for an excellent rebound of balls. Also, check if they are lined with canvas for an added layer of cushioning.
Pockets. Depending upon the type of pool table you want, look for:
Drop pockets. These are net pockets from which you have to manually collect the potted balls. They come with the smaller pool tables and the portable models.
Ball return. It is a system of sloping tracks and gutters under the playing area. It rolls the potted balls from all the six holes to a central collection point, which is often a cabinet under the slate-top. This system is installed in the larger pool tables.
Eco-friendly wood. If you are opting for a wood-dominated pool table, it is a good idea to check out whether the wood is responsibly sourced. This means that the wood used in the pool table is sourced from responsibly managed forests that ensure environmental, social, and economic balance. For this, there is a Forest Stewardship Council FSC US National Standard for forest management certification. This compliance will help conserve the forests as you enjoy billiards year after year.