Without the model number of your Laptop I couldn't tell you if you need an additional adapter, or if the laptop has built in wireless.
As for the router in the basement, unless your floors are cement, the router's wireless should reach. Somewhere in the center of where you want access would be best, so if you want it all over, then in the middle of the basement as close to the ceiling as you can place it.
The model is Inspirion 15 and it has an intel wifi and wireless n card.
The floor of my basement is cement but the ceiling, which is the second floor of my house, is not cement.