Make a website or portfolio of some of your cartoons. This is a great way to show off your work to a potential publisher.
Get a copy of "Standard Periodical Directory" and/or the "Artist's and Graphic Designer's Market." These are annual publications that list information on where and how you can submit your work. If you don't want to buy a copy, check with your local library, or ask if they can get it through inter-library loan (a system where libraries lend out materials to other libraries around the country).
Search the Internet for specific submission instructions by typing in "cartoon submission guidelines" into a search engine. Put quotes around your search criteria to get results that have all these words. This will lead you to a variety of publications' websites and will inform you about what you need to do to submit your cartoons. If you have a specific publisher or publication in mind, include that in your search criteria. Example: "New Yorker cartoon submission guidelines."
Read over the submission guidelines carefully and assemble your work in the way they request. If they want to see 12 samples, get 12 samples together. If they ask for certain information, such as your background as a cartoonist, be sure to include it. Whatever their requirements, follow them exactly to increase your chances of getting your work accepted.
Submit your work. Whether through email or through a hard copy, make sure your submission is neat and professional. Thank them for their time. Most publishers will contact you if they're interested so there's no need to send follow-up emails or letters.