Many criteria have been proposed to qualify the different properties of various election methods.
I would like to add one in light of fairly recent events: the Pencil and Paper Criterion.
The definition is simple: an election method satisfies this criterion if an election can easily be held (i.e. votes cast and tallied) without the use of any electronic device, beyond, possibly, a simple pocket calculator to make additions.
The purpose of such a criteria is to ensure the verifiability of the fairness of an election by any voter who might happen to be interested to do so.
The current use of electronic voting machines which are closed sourced and controlled by private interests is obviously a disgrace. But even in the best case scenario, i.e. with open source hardware and software, their use would take the power to verify the accuracy of the tallying of the ballots away from the ordinary people who would have no other choice to rely on the expert opinion of a minority group within the population: the technical experts who have had actual access to the machines used and who have certified them for public elections.
Never mind how many checks are made and how many guarantees are provided, if a vocal minority of people have reasons to distrust a government, the climate of suspicion will spoil the apparent (and maybe actual) fairness of the election. This is not an idle thought. Such situations regularly happen, notably recently in the US regarding the 2000 and 2004 elections.
It is still possible to cheat in elections conducted on paper only, but cheating on a large scale is much, much harder. Also, I have witnessed the satisfaction with which parents could take their children to witness the counting of the ballots: the whole process is as straightforward as it is open, so that a six year old can be explained how it all works.
The pencil and paper criterion does not apply if the ballot are not secret. In elections usually by smaller groups of people where the ballots are open, i.e. where it is publicly known who has voted what, then there can be no argument about the fairness of the election: if anyone has any doubt, they can verify that their own ballot is present and has been properly counted. They can tally the whole lot of ballots if they so desire. If anyone's ballot is missing, the voter could easily notice it and alert the electoral officer.
However the secrecy of the ballot is still sacrosanct for large scale national elections (for good reasons, unfortunately), and even if the exact listing of ballots is published (without names), it wouldn't help much an individual voter if their vote has been duly counted if the only thing he knows is that tens or hundreds of thousands of others have cast the same ballot as they did themselves.
Plurality voting satisfies this criterion. It is maybe the only quality that this damned election method may have. In many European countries, including France, elections are regularly held where voters select a piece of paper with the name of their chosen candidate on it. Later, the pieces of paper are sorted and counted. Easy.
Approval voting certainly satisfies this criterion. We are still dealing with a simple addition: 1 + 1 + 1 ... for each candidate.
The various variations of Range Voting probably could also qualify, depending maybe on the range. Given a cleverly designed tally sheet, an Emocracy election could be as easily counted as the very simple Approval voting election.
On the other hand, this criterion disqualifies Condorcet voting. I used to be a big fan of Condorcet which, indeed, has many qualities. But the pencil and paper criterion is primordial to me if we want to secure fair elections where suspicions do not prevail. Condorcet ballots could be tallied by hand, but imagine a large election with tens of millions of voters and over twenty candidates and the whole process becomes so painstaking and error prone that computers are going to be required even before such an election is held.
Besides its other known flaws Instant Runoff Voting would not qualify either. When promoted by third party candidates on the campaign trail, IRV is made to sound simple to count (pile the ballots as per the first choice, discard the smaller pile and redistribute according to the second choice, etc.). However, they take for example an IRV poll with only three ranked choices, probably the worse type of IRV voting. Once the principle of using IRV is agreed, people would want to have more ranking choices, up to being able to rank all the available candidates. Then we run again into the same dilema as for Condorcet: the ballots are theoretically easily counted by hand, but in practice, in large elections with many candidates, the use of electronic ballots is going to be required.