It's current that drains the battery.
The resistor is wasting power (1) but without it, you'll get more current, pulling more total power/energy out of the battery and draining it faster, and possibly burning-out the LED.
Current "flows" (sort-of like water). The same current flows through the resistor and LED. And with series components, the voltage divides (so you can have 7V across the resistor and 2V across the LED).
High resistance is like a skinny pipe that restricts flow. resistance w (or no resistance) resistance is like a fat pipe and the electrons flow more easily.
Resistance is "the resistance to current flow".
Voltage is like water pressure... With more voltage you get more current (through the same resistance).
This is all described mathematically by Ohms Law: Current = Voltage / Resistance.
But the water analogy only goes so far... If you cut a pipe you get zero resistance and water flows-out all over the place. If you cut a wire you have infinite resistance (no connection) and no current flows.
No water resistance isn't really a problem (as long as you don't have a flood) but zero electrical resistance is a "short circuit" and you get excess current and your battery dies quickly or your power supply blows a fuse or burns-out.
You can have 7V across the resistor, but there are no "7V resistors". LEDs do have a "forward operating voltage" and they are non-linear (their resistance goes up as voltage goes down) so if the current is properly controlled/limited the voltage "magically falls into place".
A relay is an electrically-controlled and electrically-isolated switch. For example, a relay can be used to turn on/off AC power with the Arduino which can't directly-power much more than a little LED. In fact, it can't directly-power a relay coil and it needs a driver circuit, but the relay isolates the Arduino and you from lethal voltages.**
LEDs always require a resistor or some kind of current limiting/control because they are "odd" non-linear devices. Most "things" work with "constant voltage" but LEDs want "constant current".
(1) Power (wattage) is calculated as Voltage x Current. With 20mA through the LED and 2V across it, the LED is using 40mW. With 7V across the LED, it's dissipating 140mW (as heat).