Why do we have Mirrors?

Quite simply, we have mirrors because ALL vehicles have what is know as
BLIND SPOTS!
Blind Spots are the areas around the car coloured in red that a driver would not be able to see anything within these areas just by turning their head.
To be able to see in these areas, not only may the driver need to move their head forward or backward, left or right to look around parts of the vehicles framework, but with the addition of mirrors and by using them at the correct time, it will allow the driver additional areas where they can see that would otherwise be restricted.

By introducing the mirrors, the areas colored in green have improved the drivers visibility around the car substantially.

Before moving off from the side of the road

If we moved off from the side of the road without first knowing what is going on all around us, we face the possibility of moving out into ANYTHING!
Just think of some of the consequences of NOT LOOKING FIRST
Are there any vehicle coming up from behind you?
Are there any vehicles alongside you?
Are there any pedestrians walking towards you, alongside you, or coming up from behind you that may be crossing the road or are about to cross the road?
Are there any vehicle coming towards you that may be on your side of the road overtaking a parked vehicle on their side of the road?
Is there a cyclist or motorbike passing you?
Is there a cyclist on the pavement coming down your inside that might think you are parked and decides to get on to the road right infront of you just before you move off?
Is there a vehicle coming towards you that thinks you are a parked car and wants to pull up in front of you on your side of the road?

This list isn't endless

But if you didn't look around you before moving off, any one of these situations could arise, and you wouldn't know it was happening because you didn't look first.

Before Speeding Up

Why should it matter that you check you mirrors before speeding up? Surely that won't cause a problem right?
Wrong!
Think about what the possible outcome could be.
This is the scenario.
You are merrily minding your own business driving along the road and you see a change of speed limit up ahead.
Let's say from a 40mph limit to a national speed limit.
The road ahead of you is nice and straight.
There are cars in the distance that you can see coming towards you and you decide to pick up the pace to get to 60mph

BUT! You didn't check your rear view or right door mirror before doing so.

What are the consequences of not doing so?

Just because you decided to pick up the pace, did the vehicle behind you also do the same, but quicker than you at doing it?

Have they seen that lovely straight section of road ahead of you and thought "I can get past this guy in front of me before those oncoming cars get to us?"

So you're accelerating, the car behind you is accelerating, you've not seen him having a go at overtaking you, and there are cars coming at you at potentially 60mph!!!!!

That means the closing speeds between you and the oncoming vehicles could be up to 120mph!
It isn't going to take long for you and the oncoming vehicles to meet each other. 

But you've still got this car alongside youand you are totally unaware that they are there because you didn't check your mirrors first.

You have 3 options here.
Keep accelerating, totally oblivious to the fact there is someone alongside you and hope that they brake, slow down and pull back in behind you.

A possibility that the vehicle next to you thinks you're having a race and continues to try completing the overtake, gets it wrong and there's a minimum 3 vehicle collision.

OR YOU DO THE CORRECT THING AND CHECK YOUR MIRRORS FIRST.
See that the vehicle behind you is preparing for an overtake , ease off the accelerator and just let them pass.

THATS THE SAFE OPTION ALWAYS!

Before Slowing Down

So you're travelling along the road and just want to slow down. You come off the accelerator pedal and begin to slow.
All of a sudden you feel a bump to the rear of your car, and then look in your mirror.
The vehicle behind you didn't realise you were slowing down, because just removing your foot from the accelerator did not put your brake lights on, they were sitting a little too close to you and couldn't react quick enough to your slowing and bump in to you.

However, it might be a case that you're driving along a road and decide you want to take the next turning to the right or left, you don't bother checking your mirrors, come off the accelerator and apply the footbrake and again a BUMP from behind.
Why did this happen? you applied the brake pedal which would make the brake lights come on, surely the car behind should have seen them and reacted?
The same thing applies though, HOW CLOSE WAS THE CAR TO YOU?

IF YOU DON'T LOOK BEFORE SLOWING DOWN YOU WILL NEVER KNOW.

Had you looked in your mirrors first, you would have seen there was a vehicle behind you and then been able to apply the footbrake earlier and gentler to allow the following vehicle to realise you were slowing and give them time to back off before hitting you.

Just that tiny little task of checking your mirrors first before slowing down would have led to no insurance claim, no wasted time with your vehicle in a garage being repaired, and no inconvenience of not having a vehicle whilst it was being repaired.

Always remember, 99% of the time, a vehicle that hits you from behind is usually to blame in an insurance claim. However, this could still affect your insurance premium at renewal time because you may be deemed " a risk. And that "risk" is possibly a consideration from the insurer on the lines of "DO THEY CHECK THEIR MIRRORS"
If you regularly DO NOT check your mirrors, then of course you are a risky driver and deserve to have higher insurance premiums.

Before Changing Direction

SORRY MATE I DIDN'T SEE YOU!

Well quite obviously you didn't see them because you didn't check your mirrors.

Just think about turning left into a junction.
You don't check your left door mirror and turn into the road.
Part way through the turn, you hear an almighty crunch followed by a cyclist flying through the air going over you car bonnet!

Maybe it's a turn right?
Again you don't check you right door mirror and as you turn into the road you hear an even bigger crunch, followed by a motorcylist flying through the air!
By all accounts, that motorcyclist is probably going to travel one hell of a lot further down the road than the cyclist due to higher speeds and the momentum the motorbike hits your car and the slighshot effect as their motorbike propells them through the air!

Even the basic overtake of a parked vehicle requires you to check the relevant door mirror of the direction you are going to travel in.
No point checking the left mirror if your car is moving to the right.

It could be a bike, car, motorbike, lorry, bus coach or anything that may be overtaking you. 
again just one simple glance to a door mirror in the direction you are about to move, could be the difference between a safe change of direction or killing someone.

And Regularly whilst driving along to check for changes around you

Regular checks in your rear view mirror whilst driving along will help alleviate a vast majority of potential hazards around us whilst we drive.

If you get into the habit of noticing what vehicle is following you at any given time and checking regularly, then when you check again and they are not there, you'll want to know where they have gone.
Are they overtaking you?
Have they pulled in?
Have they turned into a side road?

If you are checking your mirrors regularly enough ( without causing you to take your focus off the road ahead for too long) then you should see the vehicle following you turn into a side road, pull in, or know they are overtaking you.

These regualr checks are brisk checks just to see if anything has changed. If it has, then log whats now behind you instead.