Argumentum a fortiori is a form of argument or a rhetorical device in which one claims that since we have great confidence in one claim, we can also have great confidence in a related claim. The Latin phrase is generally translated into English as 'argument from strength', or more descriptively, 'argument from a yet stronger reason'.
"If ye then, being evil, know how to give good gifts unto your children, how much more shall your Father which is in heaven give good things to them that ask him?" -- Matthew 7:11
In this quote, the claim is that if you know how to do good, then God clearly knows how to do quite a bit better! Obviously, the target audience for this quote doesn't need to be told that God is better than a sinner; this is just a rhetorical flourish. However, if you wish to be more stringent about such things, and many people do, the argumentum a fortiori can be divided into two sub-types: a maiore ad minus, 'from greater to smaller'; and a minore ad maius, 'from smaller to greater'.
In an a maiore ad minus inference, one generalizes from a claim about a general class (or a stronger entity, greater quantity, etc.) to a claim about a specific member of that class (or etc.):
"Bob likes all kinds seafood; I'm sure he'll like oysters."
"If murder is wrong, then it follows that the death penalty is also wrong."
In an a minore ad maius inference, one generalizes from a claim about a specific case (or a weaker entity, lesser quantity, etc.) to a claim about a the general class (or etc.):
"If I can make it here, I can make it anywhere!"
"Mikey hates everything, so if Mikey likes this cereal, then surely anyone will!"
The phrase a fortiori (pronounced a-FOR-tee-OR-ee) is often used to identify that the speaker is using a form of argumentum a fortiori:
"I do not feel that I and my fellow citizens have a religious duty to sacrifice our lives in war on behalf of our own state, and, a fortiori, I do not feel that we have an obligation or a right to kill and maim citizens of other states or to devastate their land." -- Arnold Toynbee
While this sort of argument is generally considered to be acceptable, it is not very stringent. Generally, if you are writing out an argumentum a fortiori in formal logic, you will find a number of hidden assumptions that will have to be made explicit. For example, "If murder is wrong, then it follows that the death penalty is also wrong" would expand to something like "if in every case it is morally wrong to cause the death of a human being, then it is wrong to have a punishment in a legal system that entails the death of any one person". While there is nothing logically incorrect with either form of this statement, most people would evaluate these statements as being substantially different.
This difference in perception is largely because this sort of argument is used in rhetoric, rather than in logic. We can safely assume that in most debates all parties have a pretty good idea of what the other party means. In logic, one does not assume this. In the case given above, it is understood that the person speaking is attempting to frame the issue in a specific way -- that there is no moral difference between murder and execution. While this framing is not obviously wrong, there are many who will reject this framing.
Informal argument forms of this sort are very common, and many people are more than willing to hide fallacies behind cromulent rhetorical forms. As you may have noticed, it does not logically follow that just because Mikey is picky you will automatically like what he likes. Nor does it follow that 'execution' and 'murder' are the same thing, or that if you can make it in New York City, you can make it anywhere. Argumentum a fortiori is an attempt to link two ideas together, and it is easy to accept that because a link is identified it must be true. A smart audience will take the time to examine the link, and be ready to throw it out if it is not sufficiently fortis.