Necessary and sufficient conditions are ideas used to establish causal relations in qualitative research – they are one way to talk about causality when you are not using statistics or experiments.
Necessary conditions are factors that are required (necessary) to produce an outcome. For example, you could say that improving rule of law is necessary to reduce corruption – you cannot reduce corruption without it.
Another way to think about this is in reference to concepts and definitions. We can say that elections are a necessary element of democracy. If you do not have elections, then you do not have a democracy.
That is how you should think about necessary conditions, if they are absent, the outcome is impossible; or: if there is no X, then there is no Y.
Another way to think about necessary conditions is as a hoop test. A country needs to meet certain criteria – jump through certain hoops – to be considered a democracy.
Sufficient conditions are those that, if present, mean you will always see a certain outcome. If X, then Y. Sometimes, this is called a “smoking gun” – a sign that your outcome is always there. Sufficient conditions are very rare. It is more common to say that something is NOT a sufficient condition, as in, Elections are not a sufficient condition for democracy – democracy requires more than just a vote.