What is the meaning of "trapdoor" in cryptography? I do not really understand the meaning of a "trapdoor" in cryptography, so here are my questions:


What is the meaning of trapdoor and how can I convert a word or string using a trapdoor in cryptography? 
Is a hash function considered to be a trapdoor function, or is a trapdoor function some kind of encryption? 
What is the result of trapdoor function, a numeric or a string result? 
What does trapdoor itself mean? Does it mean a private key is used in a trapdoor function or does it point to the string after transmission?


It would be nice if you could explain with examples.
A trapdoor function is a function that is easy to perform one way, but has a secret that is required to perform the inverse calculation efficiently. That is, if $f$ is a trapdoor function, then $y = f(x)$ is easy to compute, but $x = f^{-1}(y)$ is hard to compute without some special knowledge $k$. Given $k$, then it is easy to compute $y = f^{-1}(x, k)$. The Wikipedia article also contains a straight-forward explanation.

The analogy to a "trapdoor" is something like this: It's easy to fall through a trapdoor, but it's very hard to climb back out and get to where you started unless you have a ladder.

A hash function is not a trapdoor function because it is not reversible. Instead, it's called a one-way function. A one-way function is similar to a trapdoor function in that it's easy to compute both and it's very hard to reverse both, but there is no special key that allows you to reverse the one-way function.

To compare the definitions of one-way and trapdoor functions, I double-checked teaching sources I'm familiar with, including:


Boneh's lecture notes
Wagner's lecture notes
Bellare-Rogaway's lecture notes
The Handbook of Applied Cryptography (PK systems, hash functions)


They all categorize (sometimes implicitly) trapdoor functions as a subset of one-way functions: Trapdoors are a one-way function with the extra restriction that they have a secret for calculating the inverse.

For example: RSA is a one-way trapdoor function, but SHA-1 is just a one-way hash function. Discrete exponentiation is also (supposedly) just a one-way function, since there's no key information about the group that can be used to calculate the discrete logarithm efficiently.

That said, on a pragmatic note I have noticed that common Internet usage often mixes the terms "one-way" and "trapdoor" interchangeably, which no doubt causes confusion.