Before they begin to use a table and chair, your child should have mastered neck and head control, and should also be able to sit upright without assistance.
In most cases, you'll find that the lowest age that a table and chair set is recommended for is around 12 months. Even then, you'll want to consider other factors, like how low to the ground the chair is, and if the table is too tall.
Like anything made for growing children, your child will eventually get too big to use most of the tables on our list. You should phase a set out once your child exceeds the maximum weight capacity for the chairs, or if the table or chair is showing signs of wear. For example, peeling paint or chipped wood can pose an injury risk, so if you notice those, you'll want to replace your table set.