Your baby starts to physically move by about 10 weeks of the pregnancy, once all their large muscles have developed, forming a thickened layer of padding between their bones and skin. Typically, you will not feel your baby's movements this early. Sometimes these movements will be witnessed during an early ultrasound. Once the baby has grown strong emough, you should begin to actually feel movement sometime between 16 and 26 weeks.
If this is your first pregnancy, it may be more difficult to feel your baby move at first. In subsequent pregnancies, many women are more able to distinguish these movements from the similar feelings of "butterflies" or "small bubbles" in your belly. After a couple of weeks of infrequent and sporadic movement, the sensations should become stronger and resemble definite kicks and movements.
Your build and size may affect your ability to feel your baby's movements. When you have a fine layer of fat over your belly, the nerves in your skin are closer to your baby. If that layer is thicker, this may cause a delay in the sensation of movements.
Your partner and others will likely not be able to sense the baby moving by touching your belly until 2 or 4 weeks after you sense your baby moving. If you are more than 22 weeks pregnant or so and feel that your baby is active, a neat trick is to place a light object such as a paper napkin on your belly while laying down. As your baby kicks the object will move, delighting and amazing onlookers.