There are several types of vinyl tile adhesives in use today. Some are multipurpose and can be used for different kinds of adhesive, and some depend on the type of tile or adhesive. Chemical adhesive removers can be purchased at hardware stores, and there are other, more environmentally friendly alternatives.
Sometimes a simple scraper will do the trick, but to get the job done without the use of harsh chemicals, even scrapers sometimes need a little help. In the case of vinyl adhesive, that help comes from putting on the heat. Most adhesives will soften when heat, perhaps through boiling water or even a hair dryer, is applied. Loosening the bond with heat allows you to gently scrape or wipe off the adhesive. It also depends on what is underneath the tiles; boiling water works on concrete but will damage wood. Use safety precautions to make sure you don't burn yourself or damage your flooring.
If water and scraping don't do the trick, there are several types of natural adhesive solvents on the market that break down different kinds of adhesive. These are biodegradable and nontoxic. If all else fails, harsher chemical adhesive removers and strippers can be used. They can release dangerous fumes, so use a protective mask and make sure the room is well ventilated.
One important thing to note before doing any kind of removal work is to check that your tiles do not contain asbestos. Asbestos fibers are dangerous to breathe if they are released into the air, and tiles that contain asbestos should be removed by a professional to avoid potential health hazards.