There are several relations between aperture and sharpness.
First one is the Depth of Field (DOF). At wide aperture (small f value), the DOF is smaller/shallower, which means only objects at focus distance will be in focus, everything closer or farther away will be blurred. Often that is a desired effect, that's why people like lenses with a wide max aperture. Keep in mind that there are other factors that affect the DOF, distance and focal length.

Another relation is that it is obviously more difficult to build a lens that at wide apertures produces sharp images (for everything inside the DOF). That is due to the nature of optics. That is why lenses with wide max apertures (small f-values) and great sharpness are typically heavier, bigger and more expensive than lenses with less wide apertures. For example, the 70-200mm f2.8 IS is much heavier, bigger, and more expensive than the 70-200mm f4 IS.

Most if not all lenses will produce sharper images if you use them not completely wide open, but stopped down 1 stop or two. However, if you stop down too much, diffraction kicks in and the image becomes less sharp. Depending on the sensor that will happen at f-values around 8 to 11 or higher.