A patch antenna is a metal or other conductor patched onto a flat surface. On the other side of the substrate, there may or may not be a ground plane. The height, form, and location of the feed line affect directionality, gain, polarization, and impedance. A patch antenna is used by most GPSs. They don't have to be rectangular; they can also be polarized in a circular fashion.
A slot antenna is a hole in a flat piece of metal or other conductor that serves as an antenna. It's similar to a dipole in that it has a complementary version. A dipole slot antenna is normally a 1/2 wavelength long slot in a wide ground plane, rather than a 1/2 wavelength of center fed wire in free space that makes up a dipole antenna.
It's possible to feed it in a variety of ways. Placing slots in a waveguide is a popular method. A set of slots may be used as a directional and powerful antenna if they are placed and spaced correctly. It may also be a slot in a PCB's ground plane, with the signal fed to the middle sides, which are always a little off center to match impedances. A dipole antenna's polarization is the same as the components' orientation, while a slot antenna's polarization is 90 degrees from the slots' orientation. It can also be fed with a 1/4 wavelength stub, much like a dipole, to balance impedance.
In the general case, the electrodynamical properties of the Microstrip slot antenna and Microstrip patch antenna are similar by the similar way of feed. On the other hand, the Microstrip patch antenna has more degrees of freedom to change its electrodynamical properties such as using different values of the dielectric permittivity of the substrate, using metamaterials as a substrate or as the passive environment of the patch antenna. The patch antenna can be used multilayer structure as well.
A microstrip slot antenna radiates as a magnetic dipole antenna. It can be fed with microstrip on the reverse side, with either a short or an open stub, as well as in other ways described by other contributors..
A microstrip patch antenna is as described by the other contributors. If it is on a substrate (or air) over a ground, then it mainly radiates from the two opposite edges, each of which looks like a slot antenna, which is a magnetic dipole.
The difference between a magnetic dipole (a slot or loop antenna) and an electric dipole (wire dipole) is in the direction of the electric and magnetic fields relative to the axis of the dipole. In the electric dipole the magnetic fields go round the dipole, in a magnetic dipole the electric fields go round the dipole. Neither of them radiate in the direction of the axis of the dipole, and they radiate most in the plane perpendicular to the axis.