I am 26 yrs old. Even though my BMI is 20, I look lean. I just want to increase bone density or size by doing physical exercise to prevent loss of bone density when I reach old age.
Load-bearing and high-impact exercise increase bone density. So squatting, deadlifting, and pressing heavy weights overhead would help. (As they say, barbell training is big medicine.) So would running, jumping, and all the variants thereof. Field sports are another fine way to make sure you give the body some impact.
Good question. You are right to be addressing bone density now. @Dave has a good answer so I will just add that a weight vest is a good way to improve bone density especially of the spine.
Assumming that you are healthy and have a healthy calcium and vitamin D intake, your bone's density and strength responds to applied stress. So as Dave says, resistance exercises (bodyweight, weight lifting or resistance bands) and weight bearing with impact (running, jumping, plyometrics etc) stress the bone and cause it to lay down new bone in the area stressed.
The National Osteoporosis Foundation has a nice pdf about preventing osteoporosis (thinning of the bones) that includes the daily amount of calcium/vitamin D you need and foods rich in calcium such as low fat yogurt and sardines. It also addresses risks for getting osteoporosis such as family history, hormones, smoking, alcohol intake, diseases and medicines that may contribute to loss of bone density.