Series: one single path
When components sit one after another on a single loop, they are in series. There is only one path, so the same current flows through every element — charge has nowhere else to go. Each resistor takes its own slice of the voltage, and those slices add up to the total supplied.
Resistors in series simply add — the total is always larger than the biggest single resistor.
This is why old fairy lights all went dark when one bulb blew: the bulbs were wired in series, so breaking one link opened the whole loop and stopped the single shared current.
Parallel: several paths at once
When components each bridge the same two points, they are in parallel. Now there are several paths, so the current splits among them — but every branch feels the same voltage, because they all connect the same two nodes. The branch currents add up to the total.
For resistors in parallel the reciprocals add; the equivalent resistance is always smaller than the smallest branch.
EMF: what a battery really supplies
A battery's job is to keep the potential 'hill' from flattening out — to lift charges from the low-potential terminal back up to the high one, again and again, using stored chemical energy. The energy it gives per unit charge is its electromotive force, or EMF, symbol \varepsilon, measured in volts.
EMF is energy supplied per unit charge. Despite the name it is not a force — it is an energy-per-charge, measured in volts.
Internal resistance and terminal voltage
No real battery is perfect. Its own chemicals and electrodes offer some internal resistance r, sitting in series inside the battery. When current flows, some of the EMF is 'used up' pushing charge through r, so the voltage you actually measure at the terminals — the terminal voltage — sags below the EMF.
Terminal voltage falls below the EMF by the drop Ir across the internal resistance; the more current you draw, the more it sags.
This is why a car's headlights dim for a moment when you crank the engine: the starter motor momentarily draws hundreds of amperes, and the huge Ir drop pulls the terminal voltage far down until the engine catches. A fresh battery has small r; an old or cold one has large r and struggles under load.