It is recommended that you use grippers for most carpets that use an underlay. You can lay carpet without grippers, but it won’t retain its shape and lay as well as a properly stretched carpet.
You can use a strong double-sided tape for underlay and carpet but this isn’t an ideal solution. It may work for a while especially if the carpet isn’t walked on frequently.
Carpet tiles do not use grippers; they use an adhesive to stick to the floor. You can buy carpet tiles that are self adhesive for quicker installation. But these will need to be laid on a sound surface. So if laying on floorboards it's a good choice to put down hardboard first to create a smooth, flat surface.