Record rough demos of your two best songs. These don't have to be a state-of-the-art recordings; a simple solo performance captured on a portable recorder will be fine for now. Rather than being intended for the ears of record company executives, these recordings are for you to study and critique.
Listen to your songs and honestly evaluate them. Try to separate yourself from the fact that they're your songs; forget all the work you've put into them, and ask yourself if you'd be impressed with them if you were a talent scout at a record company. Study their structure and arrangement. Your songs may only get 30 seconds of review time, and you need to make a favorable impression quickly, so tweak the arrangements until you make the most impact with the songs you have.
Rehearse vocal and instrumental parts until you can play and sing them with confidence. If you have a band, rent a rehearsal studio and play the songs until you can execute them with precision. There's so much competition in the songwriting field and under-rehearsed and sloppily played demos are unlikely to help your songs stand out among your rivals.
Book studio time at a professional recording studio. Your music has to stand out among the competition, and while top-notch audio quality alone won't translate into a song deal, a song with potential might be overlooked if it's not up-to-par sonically. Studio time is expensive, but if you put in the pre-production work described here, you should be able to get in and out of the studio quickly and keep the bills down.
Assemble a basic press kit to send with your submission. Keep this brief and professional. Include a biography, outlining your experience in the business, and a bulleted list of notable achievements, such as prestigious venues you've played or air-time on the radio. Add a couple of relevant testimonial quotes or music review clips along with a CD and accompanying lyric sheet, making sure everything you send has your contact information on it.
Research record companies in your genre. You can easily research these labels using an online database (see below for linked example). Stick to independent labels; the majors have neither the time nor inclination to review demo submissions unless it's via a recommendation; unless you have a contact within the label, don't waste your time. Review the company's submission guidelines on its website and address your press kit directly to the contact person, usually the A & R (Artists and Repertoire) representative or talent scout.
Email the representative and ask if you can send your music. Mail out copies of your demo and press kit to any companies that respond favorably, marking the envelope "Requested Material." Make sure that you address it directly to the person you corresponded with, and after a week, send a follow-up email asking if the packet was received. If the company decides to pass on it, don't pester the representative for a detailed critique. Be professional and thank him for his time. This will hold you in good stead should you wish to submit new music in the future.