What happens to a computer's power consumption when it goes to sleep?

It consumes less.

Sleep mode powers down everything except for a mere trickle of power to RAM, where your active state is preserved. Your computer’s RAM requires an ongoing current of electricity or else it will reset.

When you wake the machine back up, it “remembers” everything and so needs not go through the boot up process all over again.

“Sleep” is conceptually different than “hibernate” (although manufacturers can sometimes blur the terms). Typically, hibernation is good for much longer period of time, nigh indefinitely. The contents of RAM are written to non-volatile disk where they can stay practically forever. When the computer wakes up, it reloads the RAM with the information stored on the disk.

Waking from hibernation takes a bit longer than waking from sleep, however both are significantly faster than the typical boot cycle.

It goes on vacation.

If you really want to know what exactly your computer’s power consumption is when on, in standby, hibernate, sleep, whatever, and off but plugged in, get one of these

@robmandu has it right except for one thing. While it’s true that booting a Windows system takes longer than waking up from sleep/hibernation, my experience with Linux is that it takes about the same amount of time, so it depends.
However, since most people run Windows, then what he says is usually true.

Personally, I tend to skip Sleep mode entirely on my laptop and either hibernate it or shut it down entirely. Leaving a laptop in Sleep mode will draw the battery down but hibernation has zero power draw and can be maintained indefinitely, and you can remove the battery (or, for a desktop system, survive a power outage) without losing data.

On a mac laptop, you can put it to sleep just by putting the cover down. I don’t know how to set it to hibernate mode though. Or maybe you can’t set mine, cause it’s 7 yrs old.

@28lorelei PC laptops can be set to sleep when the lid is closed too. In fact, it’s the default behavior unless you go into the Control Panel to change it. As for Hibernate, that is a relatively new thing. PCs have only had that ability since WinXP, and some of your older hardware does not support it.
Macs do not really have a “Hibernate” mode though some of the newer Macs can be put into hibernation with either technical wizardry or third-party software.