Will cell phone SIMS cards fit other brands of cell phones?

In other words, are cell phone SIMS cards interchangeable in different brands of phones?
I’m not so comfortable with Casey’s link.
Can cell phone sims cards of the same make be switched?

My cellphone got soaked with water and quit, it was an inexpensive prepaid(Tracphone-Motorola)one, but I’d like my phonebook and number back. If I buy the same brand and make phone and put the old sims card into the new phone, will I have my old phonebook and number?I’m jobhunting and gave this # on several applications.