Did I need a stronger Drill?
Perhaps not, often you can just lubricate the screws before driving them. People suggest soap, I've sometimes dipped screws in a light grease (which can leave marks on the wood). This really makes a huge difference.
Another alternative is to drill small pilot holes before inserting the screws. The width of the drill needs to be less than the full width of the screw threads.
If you are driving a lot of screws and wish to avoid these extra steps, I'd buy a stronger drill/driver.
What type of drill would be advisable
You might own a Bosch 7.2V screwdriver (model PSR7.2-LI) that has a torque (turning strength) of 6 or 10 Nm.
A 10.8V Bosch GSR 10,8 V-EC HX has a torque of 20 or 18 Nm.
A 14.4 V PSR 1440LI-2 has a torque of 17 or 34 Nm.
(The two ratings are usually for hard and soft materials)
You could buy a tool with a higher torque rating. Usually this is associated with a higher voltage rating (for battery powered tools). Bosch always provide torque ratings for this type of tool.
Remember that higher voltage tools are usually bulkier and heavier. smaller tools are often more convenient in confined spaces and less tiring for extended use.
It is usually advisable to buy a driver with an adjustable clutch (a turnable ring with numbers) that allows you to reduce the torque to avoid driving the screws too deeply (or from damaging the screws). Not all electric screwdrivers have this as standard but most do.
Emma's answer lists the basic types of tool available for this task. You don't need to drill into masonry so it is useful to distinguish between an axial hammer-action drill (mainly for masonry, but not especially good at that unless you buy a mains-powered or SDS-plus drill) and a rotary impact driver (advertised as good for driving screws into wood). See When would one use an impact driver versus a regular drill?