TM stands for "Technical Machine". They can be used to teach your Pokemon new moves. Both Meowth and Persian are Normal Type, and a part of the Field Egg Group. That means that they can learn more TM moves than most other Pokemon can. Normal type Pokemon are known for that, and some trainers use it to their advantage...

To give you an example, both Rich and Alice here know the move Thunderbolt after being taught the corresponding TM. It's not a move they can learn naturally, but now that they know it, any eggs that they breed will know the move as well. Eggs that have TM moves bred down onto them are usually more sought after, as well.