Well, you do not actually summon your companion dragon :$ You have to astral project and then go to the astral plane. Once there, you have think of going to the dragon realm. When there, you will see many dragons, but you will know which dragon is yours, he/she will come to you and you will sense that he/she is your dragon companion. And by how it feels, it feels magnificent. You feel that you have a true friend that comforts you when you need comfort, protection when you need protection, and he/she helps you in your dragon magick. Hope this helps...watersage14
More commonly referred to as your Guardian Dragon, and you do not "summon" it either. Summoning is done to bring forth a spirit for work or to serve you. Dragon energy serves no man, but can and will work with you in your daily life, and magick practice.
Most common way of making contact is through meditation. You can visit the Dragon realm through AP but not required to find or ask to meet your Guardian/Companion. Clearing your mind of all things, and focusing on your desire to make contact is all that is required. Though some people like to speak out loud which is ok as well. Be patient, it takes time.
Throughout your practice you can come in contact with many different Dragons, each may only stick by you for varying amounts of time, each teaching you something different or building upon knowledge you already possess. As for what is like, when you do meet one you will know for sure that mankind is not alone, nor the biggest presence in the world.
Be sure to be polite and respectful. Being disrespectful at all towards any Dragon entity or energy will result in many unlikeable ways and including being shut off from their presence.