Answer
Ionotropic receptors and metabotropic receptors are the two classes of receptor proteins that neurons express to receive neurotransmitters. A receptor protein is any protein whose primary function is to transduce the binding of a molecular substrate into some signal or action. Ionotropic receptors directly pass current when they bind substrate. On the other hand, metabotropic receptors directly activate molecular cascades when they bind substrate.
Ionotropic Receptors
Ionotropic receptors are the most common neurotransmitter receptors. Because their effects are direct, they are rapid and, generally, very short-lived.
Examples abound but include the AMPA receptor, the primary receptor for the primary excitatory neurotransmitter, glutamate, the NMDA receptor, which is thought to be involved in learning and memory, and the GABA receptor, the primary receptor for the primary inhibitory neurotransmitter, GABA.
Note that NMDA is involved in learning and memory, two processes defined largely by their temporal extension. The NMDA receptor is capable of having a longer effect because the current that it passes includes calcium ions. Calcium is a common 2nd messenger in molecular cascades, so this current can have effects normally associated with such cascades, rather than with simple currents. Check out the "Aside on Terminology" below for some thoughts on what this means for the distinction between ionotropic and metabotropic receptors.
Metabotropic Receptors
Metabotropic receptors have more widely varying effects than ionotropic receptors, but they almost all fall into a single gene family: the G-protein coupled receptor family. These proteins associate with a common, smaller protein, the G-protein, and release it when they bind their substrate. It is often the case that these cascades involve the cyclic nucleotide family of signalling molecules and/or protein phosphorylation. Because they must activate a chemical cascade, metabotropic receptors have effects on a slower time scale than do ionotropic receptors. Note, however, that we've seen an example of an ionotropic receptor that can have effects that last for a very long time. Similarly, there are metabotropic receptors whose cascades merely activate ion channels, causing voltage changes on the order of a few hundred ms.
Examples of metabotropic receptors include the metabotropic glutamate receptors (mGluRs), which are involved in long-term synaptic depression, and the muscarinic acetylcholine receptors (mAChRs), whose roles in the central nervous system are poorly understood.
An Aside on Terminology
Plato famously proposed to define a human being as a "featherless biped". The cynical philosopher Diogenes as-famously responded by plucking a chicken and shouting: "Behold! I have brought you a man."
The NMDA receptor† is about as problematic for our definitions as the chicken was for Plato's. In some sense, it is both ionotropic and metabotropic: it passes a current and that current happens to directly activate a molecular cascade. This poses a problem, since we'd like to be able to split our molecules into disjoint sets.
I asked this question (is the NMDA receptor metabotropic or ionotropic?) at a neuroscience conference and a fight almost broke out in the hot tub. Everyone agreed it was a stupid question, but there was disagreement on the right answer. To biophysicists and ion channel specialists, NMDA was obviously ionotropic, while cellular and systems neuroscientists thought it equally obvious that NMDA was metabotropic.
So what's a girl to do in the face of this confusion? I am comforted by a quote from Ludwig Wittgenstein, another philosopher, this time from the 20th century. In a quote that unfortunately does little justice to his fabulous philosophy of language, he said "The borders of my language are the borders of my world". That is to say, the world itself simply exists, names, jargon, and symbols be damned. We place artificial borders on it with our words, just as nation-states place artificial borders on the Earth, and then we pretend that our categories and labels are "natural" when they are really anything but. They reflect the world we believe we live in, the world we wish to live in, and we should take care to be honest about that.
† An aside on terminology to this aside on terminology: the convention of naming receptors after the chemical that happens to be the most selective binding agent at the time of discovery is totally bonkers. There is no NMDA (nor is there AMPA!) in the body, so calling the protein in question an "NMDA receptor" cuts totally against the very notion of receptors^!
^ One level deeper: the notion of receptors seems to me to get the whole thing precisely backward. The term comes from a time when biochemistry was seriously limited in its techniques, and considerably underestimated the complexity of proteins and the power of models from inorganic chemistry. It refers to assays where chemicals are washed on and their binding affinities calculated very cleverly. Their role in the cell has little to do with binding per se. Instead of being passive receptors, these molecules are detectors. They turn signals in the cells environment into physical and chemical changes in the cell. Note that this refuses any need to get weird and say things like "photoreceptor" or "mechanoreceptor".