To save some time looking at the external references, here is my ladder logic cheat sheet.
Ladder diagrams originated in pre-computer days as a systematic way of drawing relay logic schematics.
The vertical lines on the left and right sides of the diagram represent the power supply rails: line voltage (hot) on the left side, and neutral on the right side.
The things that look like capacitor schematic symbols are normally open contacts: either PLC inputs from sensors, or contacts off of internal relays.
The things that look like capacitor schematic symbols with a slash through them are normally closed contacts: either the negation of an input sensor signal, or the negation of an internal relay state.
The thing on the right hand side immediately before the vertical neutral rail is the output for the ladder rung. It may be a physical output of the PLC, or it may be an internal relay coil.
A major difference between logic done with relays (which the PLC's emulate) and solid state logic gates (AND, OR, NOT, etc.) is that:
a) Whereas logic gates typically have multiple inputs, a relay has only one input: the relay coil. This means that you have to combine inputs, as relay contacts, in series and parallel to get the required logic function, and then feed it to the relay coil. In essence the logic function is not done in the relay itself, but in the series/parallel wiring of the contacts that control the relay.
b) Whereas a logic gate has only a single output, a relay coil has multiple outputs, both normal (N.O.) and inverted (N.C.). These multiple outputs are the basis of the series/parallel logic that control subsequent relays.
Relay ladder logic can do everything that that gate logic can do, but it requires a different way of looking at the problem. Once you're familiar with relay/ladder logic, it's very easy to follow. This is the reason why PLC vendors went with ladder logic in the first place. Electricians were very familiar with relay ladder diagrams that had been use for many decades before PLC's became available.