1 μm = 10-6m = 10-3mm
1 nm = 10-9m = 10-3μm
One can use a light microscope to see living cells, its plasma membrane, nucleus and probably some most prominent organelles (with the help of a computer), as it can not see things smaller than 0.2 μm (due to light wavelength, not to the quality of lens).
An electron microscope uses a beam of electrons to create an image with much finer resolution than a compound light microscope. Most organisms have to be killed before they can be observed in an electron microscope (the sole exception being a tick). Some transmission electron microscopes can have a resolution of better than 50 pm, and can magnify up to 10 000 000x. This is smaller than the wavelength of visible light, which is between 390 - 700 nm, so it cannot use light; thus, electrons are used rather than light. The two main types of electron microscopes are transmission electron microscopes (TEM) and scanning electron microscopes (SEM), and both of them usually need a partial vacuum in order to view the specimens, making them costly. However, they are much better for perceiving cell details than <2 000x magnification compound light microscopes.
Some very large cells can be seen with the eye alone. An example of this is gromia sphaerica which has a maximum specimen size of 38 mm (millimeters).