I am 16 years old, around 5'5, 105 lbs. and skinny, what would be a good way to gain muscle?

Try the T25 workout.

All you really need is a cheap set of elastic resistance bands.

Use the bands for biceps and deltoids , calf and hamstrings
Do push ups for almost everything else above the waist
Do lunges and squats (no resistance, just body weight) for quads and glutes.
Get rid of most of the useless carbs in your diet.
Do resistance work every other day for six months. Walk or jog the other days.
You will see a huge difference.
Plus be patient. At sixteen, it is likely you are still not producing a whole lot of the hormones that will promote increase in muscle mass. You are still sort of a little kid. Your time is coming.

Have you looked at bodybuilding.com ? They have a ton of workout routines…they fit pretty much anyone’s situation. If you’re looking for something that doesn’t require a gym pass, they have plenty of at-home workouts. And plenty beginner ones, too.
There’s so much you can use at home for working out other than your own resistance. I used to use chairs to do dips, detergent bottles filled with water and sand for dumbbells, etc. Good Luck!

Push-ups. Lots of them.

And eating is also extremely important. This is coming from someone who was in your exact position. Don’t be like me and think that the only thing you have to do to gain muscle is work out. I only started gaining weight once I started eating. This means eating healthily, not just anything. Focus your meals on proteins and carbs with some good fats. And don’t forget fruits and vegetables, either.
Look at the nutrition labels on food!

Good Carbs:
Fruits and vegetables (including potatoes and corn)
Pasta
Rice

Bad Carbs:
Candy
Chips
Cookies and pastries
Other sugary things that you probably think won’t be healthy

Good Fats:
“Unsaturated fat”: Nuts, Fish (fish is mainly protein but it has Omega-3 fat)

Bad Fats:
“Saturated Fat”

Give it at least a month. Try to get at least 2000 calories a day and don’t forget to have a protein shake right after you work out. As far as fast food goes, try to avoid it but it is better than nothing. Don’t skimp because you’re only skimping yourself. It’s not the easiest thing to do. That’s why not everyone has perfect physiques (other than genes). I had to not only cook all my food, but also find money to buy it and go shopping for it. It takes a lot of my time to do that and also exercise. It’s a big commitment but it is worth it if you really want it.

Physically demanding chores. Two birds with one stone.

Hire some.

You must follow a good diet with alot of protein and complex carbs and healthy fats. Diet is accountable for 70% of muscle growth. You need to join a gym and practice squats, deadlifts, bench press, shoulder press and pull ups/pull downs. All these help to gain muscle. Don’t overdo the workouts as your body is still growing and developing. Please don’t take any supplements at this age, because an increase in testosterone production at this point will damage your body. I don’t think there is harm in taking extra protein.

A great simple workout for arma can be achieved by using ½ litre waterbottles as dumbells- sounds silly, but its agreat start , and you can up the weight by filling them increasingly with sand or gravel as well as/ instead of water. As a rough guide, doing more reps (repetitions) with a lower weight improves muscle tone and definition, whereas fewer reps with a higher weight increases mass. Good luck brother/Sister! :D

The diet tips already mentioned are important. I grew up on a farm, where it was easy to build up. @talljasperman is absolutely right about the chores. It may sound like an adult ploy, but it it exactly the truth. Do you know why muscles are attractive to us to begin with? It is because they tell us someone is healthy, and capable, two attributes neccessary for completing a day’s physical tasks. Chores, muscles, they go together.
When I was in the Navy, I was a skinny little woman, 5’3”, 109 lbs as I entered bootcamp. I was muscled, but thin. Bodyfat was too low to calculate.
One part of my job developed my upper body more than any other. Hauling fenders in from over the side. It involved hand over hand motion, lifting a rope with a fender hanging from it which weighed as much as I did, and then flipping it up over a waist high rail. I had to repeat this task dozens of times a day, between other heavy duties.
Remember that whatever you do to build yourself up, to be careful HOW you do it. Lift properly, so you don’t hurt your back. Don’t do too much of the same thing in the same place. Your body needs to change position and routine after a while.
Have you ever watched the Karate Kid movie? He thought he was just doing chores, but each chore was exercising just the right muscles for certain functions.
There is no shortcut to building muscles. You just as well do it by accomplishing something needed. If you don’t have chores at home to work with, look around you. There is always some person who could use help. Maybe a woman who has just had a baby, and her husband is away in the military,or a truck driver or something. Maybe you have an elderly neighbor or two who would be delighted if some youngster offered to take out their trash, or paint her fence, or dash to the corner store and pick up a few things for her. Any one of these tasks would benefit you, and at least one other. BONUS!

Steroids.

Time? It’s not unusual to be skinny at your age. You’re probably in the middle of a huge growth spurt.

Haleth has a good point.
Just think it you remain skinny in your middle age a lot of people will be envious of you!

A bodybuilding regimen can work wonders in building much muscle. Barbell exercises combined with high protein diet and adequate sleep are the way to achieve your goal. My first boyfriend gained sixty pounds of muscle in his first three years of bodybuilding. Progressive resistance is necessary for best results, so anyone who advises you to do push ups, etc. is wrong. Bodyweight exercises cannot be sufficiently progressive for fullest development of your muscles.