You can obtain a board game template from one of numerous websites and use that to print up your own blank game or add information to the spaces before printing it. Some hobby and craft shops sell the heavy cardboard that most companies use to print board games, which you can use to make your own game sturdier and easier to play.
While you can make your own game board design in a simple graphics computer program, there are also many resources online that you can use. Hobby and gaming sites often have simple, blank designs for a board game, which you can download or print directly from the site. It is easy to then add your own elements to such a game, either in a computer program or by drawing them by hand onto the spaces of the board.
Any type of paper can be used to print out a game board, but consider thick, strong paper such as cardstock to really make the board easier to use. Quite a few hobby and gaming shops sell blank, sturdy boards that are perfect for gaming, even some that fold up into sections. You can print up a game board design on any kind of paper, and then glue it onto the heavier board so that it lasts longer. Consider printing your design and laminating it before attaching it to the board, so that it can survive spills, dirt and other accidents.