I have been trying to understand what is exactly a sieve and why sieves are useful. I have read Wikipedia articles about sieve theory but they don't provide a definition of what is a sieve or why they are useful.

Can someone explain what is a sieve in general terms and what properties of sieve makes them so useful in attacking in number theoretic problems? Are there general principles that one can use as guideline to see if sieves are likely to be useful in attacking a problem?

This question might be a bit elementary for number theorists, but I wasn't able to find good and concise information at the level of a general mathematician about the topic. (The Wikipedia article links to Ben Green's notes about sieve theory but the link is not working anymore. I Googled a bit but couldn't find a concise high-level but rigorous exposition, notes like this and this dive quickly into examples and application without giving any general definition and discussion of sieves.)