Oversimplifying what P=NP would mean for dummies:
1) you have a problem known to be slow to solve(in NP)
2) you create a description of your problem as a sequence of logical operations (if you can make a logical circuit for it this can be done)
3) you translate that description in a special formulation known as CNF using just 3 variables(this point was already proven as possible and fast for all NP problems)
4) ??? - you take your newly developed polynomial time 3 SAT CNF solving algorithm and use it to compute a solution to the problem stated at 1)
5) $$$ - you now have a polynomial time algorithm to solve the problem stated at 1)
P - Polynomial time = Fast to solve even for very large inputs (finding if a given number is even)
NPC - Non determinisctic polynomial time complete = Very very slow to solve for somewhat large inputs (finding a solution for the travelling salesman problem with lots of cities)
NP - Non deterministic polynomial time = Contains all problems that are in P in addition to many others that aren't in P.
CNF - Conjunctive normal form = A way to write boolean algebra expressions(sequences of logical operations)
SAT - Satisfiability = Find out if there are any values for the variables that make a boolean expression true
P.S. for those still reading:
As for prime factorization it is known that it is in NP and in Co-NP but AFAIK there exists no proof as to whether it is in NPC, co-NPC or P. But we also know it to be in FNP and we know that if P=NP then FP=FNP and vice versa, hence prime factorization would be solvable in polynomial time by a turing machine as all FP problems are.
See your favourite boolean algebra/logics book for CNF and SAT and your favourite computation theory book for Complexity.