Sure thing! You need lumber from cutting down 4 trees. (Note: you don't get more lumber if you cut down more trees.) Then go to Town Hall and Mrs. M will give you a deed to your house. Go back outside and tap on the deed to activate it (it will glow). That will give you an image of a house. Drag the house to any unoccupied lot, then tap on the green box to accept the placement.

I chose my lot first and walked my mayor over there before activating the deed so I didn't have to drag the image of the house anywhere. The style of house you get will be random, but you can either tap the deed on and off to look at different styles before you hit the green button, or change it later by buying house upgrades from the Bank.