(See the sibling comment with a "10." in it for context.)
1. Bounded gaps between primes. This one's pretty easy to describe, though the details of how it works are harder.
There seem to be quite a lot of pairs of prime numbers just 2 apart. 3,5; 5,7; 11,13; 17,19; etc. Are there infinitely many? Everyone expects that there are, and there's even a plausible conjecture for how common they are. But no one has been able to prove it. This is yet another of those "additive versus multiplicative structure of the integers" things, like the Goldbach conjecture. And, like the Goldbach conjecture, what everyone believes is that the two structures are kinda independent, almost as if the prime numbers are chosen at random apart from some simple constraints.
Zhang made a big step in the direction of this "twin primes conjecture", though we're still a long way from proving it. He proved that there are infinitely many pairs of primes no more than 70,000,000 apart. (Other people have subsequently reduced that bound to 246. Still a lot bigger than 2.)
I don't know enough analytic number theory to say much about how Zhang's proof works. The first (and deepest) step is to get some bounds on (roughly) how much the density of primes that are k mod m can vary with k. (So, e.g., any prime number >3 must be either 1 mod 6 or 5 mod 6; if you look at prime numbers between x and 2x, how different can the number of them that are 1 mod 6 and the number that are 5 mod 6 be?)
Having done that, he uses a technique called "sieve theory", which has been around for a while, to prove that for a certain (rather large) k the following holds: If you have an "admissible" set of k numbers -- I'll say what that means in a second -- then there are infinitely many n such that when you add n to each of your k numbers, you can find two primes among the results. And "admissible" means that there's no prime for which your k numbers cover all the possible values mod p. So, e.g., {2,4,6} is not admissible because these numbers include every possible value mod 3. But {2,4,8} is admissible.
And then getting from there to the actual theorem turns out to be pretty simple.
What came next after Zhang's work was that, as I mentioned, a bunch of people whittled his bound from 70 million down to 246, and established a connection between this work and something called the Elliott-Halberstam conjecture, and found that if EH is true then the number would go from 246 to 12 -- or even to 6, if a stronger version of EH is true. It doesn't seem possible to go further than that without major new ideas.
1. Bounded gaps between primes. This one's pretty easy to describe, though the details of how it works are harder.
There seem to be quite a lot of pairs of prime numbers just 2 apart. 3,5; 5,7; 11,13; 17,19; etc. Are there infinitely many? Everyone expects that there are, and there's even a plausible conjecture for how common they are. But no one has been able to prove it. This is yet another of those "additive versus multiplicative structure of the integers" things, like the Goldbach conjecture. And, like the Goldbach conjecture, what everyone believes is that the two structures are kinda independent, almost as if the prime numbers are chosen at random apart from some simple constraints.
Zhang made a big step in the direction of this "twin primes conjecture", though we're still a long way from proving it. He proved that there are infinitely many pairs of primes no more than 70,000,000 apart. (Other people have subsequently reduced that bound to 246. Still a lot bigger than 2.)
I don't know enough analytic number theory to say much about how Zhang's proof works. The first (and deepest) step is to get some bounds on (roughly) how much the density of primes that are k mod m can vary with k. (So, e.g., any prime number >3 must be either 1 mod 6 or 5 mod 6; if you look at prime numbers between x and 2x, how different can the number of them that are 1 mod 6 and the number that are 5 mod 6 be?)
Having done that, he uses a technique called "sieve theory", which has been around for a while, to prove that for a certain (rather large) k the following holds: If you have an "admissible" set of k numbers -- I'll say what that means in a second -- then there are infinitely many n such that when you add n to each of your k numbers, you can find two primes among the results. And "admissible" means that there's no prime for which your k numbers cover all the possible values mod p. So, e.g., {2,4,6} is not admissible because these numbers include every possible value mod 3. But {2,4,8} is admissible.
And then getting from there to the actual theorem turns out to be pretty simple.
What came next after Zhang's work was that, as I mentioned, a bunch of people whittled his bound from 70 million down to 246, and established a connection between this work and something called the Elliott-Halberstam conjecture, and found that if EH is true then the number would go from 246 to 12 -- or even to 6, if a stronger version of EH is true. It doesn't seem possible to go further than that without major new ideas.