You may think that "lay down" and "stay" are two separate commands. It's actually easier to teach them as a combination. The key to success is consistency and patience. Practice for 5 to 10 minutes each day and reward your dog with treats and praise while he's learning.
Choose a quiet place where your dog can focus on you and won't be distracted. Use a special treat that you reserve for training only. Hold the treat in one hand between your thumb and forefinger so your dog can see it.
Place your dog in a sit position. Hold the treat near your dog's nose so he can smell it, but don't let him have it. Give the "down" command and slowly lower the treat between his front legs to the ground. Your dog's head will follow the treat and his body will start to lower to the ground.
Move the treat slowly away from your dog in a line across the floor. As soon as he is lying down, say "Yes!" and give him the treat, with lots of praise. Encourage him to stand and repeat the exercise.
As your dog responds to the "down" command more quickly, eliminate the treat but continue the sweeping "down" motion with your hand. Hold the treat now in your free hand but don't let the dog see it. Tell him to "stay" and back one step away. Count to 10; treat and praise.
Continue the down and stay exercise daily, extending the amount of time your dog remains in the down position until he will stay for 5 to 10 minutes. Gradually eliminate the treat and use the hand signal and verbal commands only. Choose a word such as "OK" to release your dog from the "stay" so he'll know when he can get up.