Fixed-term elections are those whose date can not be set by politicians. The US presidential election is perhaps the most prominent example of this. There are also elections, such as that for the Australian House of Representatives, which have a date effectively set by the head of government (so long as it falls within a certain number of years since the previous election).
What are the advantages and disadvantages of these two systems?
Comparing presidential systems like the US with parliamentary systems like Australia is a bit like comparing apples and oranges. So to attempt to answer the question, it might be useful to provide some context:
It could be argued that all democratically-elected positions have maximum term lengths, but that some have mechanisms that allow elections to be held earlier than originally scheduled.
So, for most parliaments, a maximum term length is specified, but a general election can be called early if the government loses a confidence vote, or (in some countries and states) the prime minister (or equivalent) requests it.
For other elections, there may be an equivalent of a confidence vote (e.g. a petition to hold a recall election), or rules that require an election to be held if an office-holder dies or resigns (as in many legislatures).
In the US, there is no equivalent of a confidence vote because the government does not sit in Congress, as is the case in parliamentary systems, so US general elections can never be held early, and hence are always fixed terms.
So to actually get back to the original question: I would interpret "fixed terms" in this sense as only applying to a parliamentary system which does not allow early general elections.
Allowing the Prime Minister to call an early election means that the government can choose a time when it is popular in the polls, rather than waiting for the end of its term, when it may not be so popular. Whether you regard this as a good thing or not depends on whether you support the current government!
Conversely, not allowing the Prime Minister to do this takes the politics out of things, and allows all parties (including the government) to plan ahead with some certainty.
Most parliaments still allow early general elections if the government loses a confidence vote. The rationale is that, if no party is able to reliably win votes in parliament, they have little chance of convincing parliament to agree to their budget, which is considered an essential part of being a government.
The advantage is that it prevents the party in power strategically choosing the election timing so that it advantages them.
Now sometimes a party lacks the numbers in parliament for the policies it wants to pass, and parliament ends up deadlocked. If the term is fixed, it prevents the party calling a new election to let the voters settle the issues.
One of the disadvantages of fixed-term elections is that since everyone knows years in advance when the election is going to be they start campaigning years in advance. These long campaigns cost a lot of money, so it is difficult for people who are not rich to stand for office. A system in which for most elections the date is only known a few months in advance means that serious campaigning only takes place during those few months. This is cheaper for the political parties and less boring for the citizens.
