Intubating dogs and cats is not only an important task and something we do almost every day, but it is critical that it is performed correctly. Here is a step-by-step description regarding intubation. 
1. First choose the tube that you will use. The tube size will largely depend on the size/body weight in dogs.

Note that this is not always accurate as there is no cookie cutter recipe for size (Yorkie’s tend to have a larger trachea than you would expect, or overweight dogs may have a smaller trachea than you are expecting, etc).  I always have one size above and one size below my chosen size ready in case I need to pick a different one once I get in there. 

Here is a chart for general guidelines in choosing a size:

2. The tip of the tube should be placed midway between the larynx and the thoracic inlet. I like to hold the tube up next to the patient to pre-measure about how far in the tube will be placed. 

3. After selecting a tube, pressure check the cuff to make sure there are no leaks and make sure the tube is clean (inflate the cuff with air using a syringe and make sure it stays inflated and no air is leaking out and then deflate it again).

4. Apply a small amount of sterile lube around the end/edges of the tube so it will slide in easier.

5. The patient should be in sternal recumbency as the induction agent is given IV. We mostly use Propofol for induction, so it is given slowly and to effect.

6. I have someone hold the head for me and the upper jaw, while I grasp the tongue with a dry gauze to open the mouth. If you use a laryngoscope, you may need the assistant to also hold the tongue out so you can use your free hand for the laryngoscope.

7. Gently press down the epiglottis with the tip of the endotracheal tube to visualize the tracheal opening. Pass the endotracheal tube through the glottis and into the trachea until tip of tube is midway between larynx and thoracic inlet (you have pre-measured this prior to induction so you should know about how far to insert). You MUST visualize the tube going into the trachea, don't try to do it blindly as this is how mistakes are made.

8. Now secure the tie gauze around the endotracheal tube and tie to patient (either behind the ears or top of muzzle, depending on your procedure).

9. Inflate the cuff, and then auscult the chest on both sides for breath sounds. Lay the patient down on their side. Be careful to sufficiently inflate the cuff without over-inflating (cats are especially at risk for tracheal injury from overinflated cuffs).

10. Connect the tube to the anesthesia machine. I normally will give a few assisted breaths once the tube is connected. You should listen carefully for any air leaking when you give the breath. If you hear air, make sure the cuff is properly inflated. The tube should always be a good fit and should never be too loose inside the trachea so no air should be escaping once the cuff is inflated.

11. After induction, I always immediately connect the pulse oximetry before I begin connecting the rest of the monitoring equipment. It is the fastest way to determine oxygenation and heart rate as you get started. Don’t forget to monitor the tube/cuff throughout the procedure. If you move the patient during the procedure (turn sides, etc.), the tube should be disconnected from the machine and then reconnected after moving. Take special care to make sure that the cuff remains inflated, and that the tube doesn’t get kinked.

12. When the procedure is completed and when the pet is disconnected from the machine, the tube is deflated. After the patient is swallowing the tube can then be removed.

Here is a great article and video from Vet Girl you should watch: How To Intubate A Cat Video from VetGirl 

Check out this video on procedure and placement of the tube into the trachea in a dog by Dr. Stephanie Parks: