The "mono" part of monoprint or monotype should give you a clue that these are printing techniques that product one-off prints. The words tend to be used interchangeably, but The Printmaking Bible distinguishes between the terms thus:
A monotype is "a singular print created through an acknowledged process that can be learnt and replicated to gain similar effects with different images" and a monoprint is "a singular work that can be produced without the need to undergo a series of steps."1
A monotype is created using a printing plate without any lines/texture on it; a unique image is made in the ink each time. A monoprint is uses a printing plate with permanent elements to it, for instance engraving lines. Although how you ink the plate produces different results, these permanent elements will appear in every print.
Call it whichever you will, the printing technique can basically be done in three ways, all of which involves either putting printing ink or paint on a non-porous surface (such as a piece of glass) and then applying pressure to transfer it to a sheet of paper. The first monoprint technique (trace monoprinting) is to roll out the ink or paint on the surface, gently place a sheet of paper on it, then press onto the sheet of paper to selectively transfer the ink to the paper and create the image by where and how you've applied pressure.
The second monoprint technique is very similar, except you create a design in the ink before you place the paper, then use a brayer (or spoon) on the back of the paper to transfer the ink. Use something absorbent such as a cotton swab (bud) to lift paint, or scratch into it with something hard such as brush handle (sgraffito).
The third monoprint technique is to create the image as you place the ink or paint on the surface, then use a brayer, back of a spoon, or printing press to transfer the image to the paper. For step-by-step demos of this technique, see How to Make a Monotype Print (very detailed demo was done using a waterbased monotype paint, which is then encouraged to "lift" from the surface by having the paper damp, not dry) or How to Make a Monoprint in 7 Steps.
What Do You Need for Monoprints? You've lots of options and should experiment to find what works best for you. Various types (and colors) of paper and whether it's totally dry or damp will give you different results for starters. You can use printing inks (oil-based inks dry slower than water-based ones, giving you more working time), oil paint, slow-drying acrylic, or watercolor/tempera with damp paper.
I use a thickish piece of plastic "glass" from a picture frame for rolling out my ink. You want something that's easy to clean, smooth, and won't break if you apply pressure to it. You don't need a brayer (though it's fun to use), you can apply the ink/paint by brush for a monoprint, with any brushmarks in it giving texture to the print.
1. The Printmaking Bible, Chronicle Books p368