What would happen if Earth's polarity was reversed?

Compasses would be seriously ****ed up.

Pole Shifts have happened more than 180 times before, as detected in the paleomagnetic record of rocks on the ocean floor and in some lava flows. These magnetic reversals take place every 100,000 to 25 million years, according to Scientific American, and may take as long as 5,000 years to do so.

We'll most likely experience aurorae at all latitudes whilst the dipolar magnetic field settles down to its new, reversed state, and there might be a small increase in energetic particles from space
we'll still be protected by our thick atmosphere.

The earth's magnetic polarity reverses itself periodically and this has been taking place since the earth's creation.

There are 2 polarities - Geographical polarity; Magnetic polarity.
1. Geometrical : That is the Earth tumbles over. After that, Sun rises in the opposite direction to what it is now, move in opposite direction (& sets all opposite). 'East' or 'West' are the names humans give and it is their convention. Apart from that everything will be same.
2. Magnetic : As per convention Compass' N pole points to a place near the N geographical pole. As per our convention No.2, that pole near Geog. N pole must be named Magnetic South pole (Unlike poles attract, so it must be S). Inside the core of Earth, the molten Iron swirls (revolves) creating its field, we call Geo-magnetism. It might 'flip'. The result will be magnetic Compass on a ship will have its N pole pointing to Antarctica (Geog. S pole).

Even if they do any of these, it will be a gradual process over some ten to hundred thousand years. Earth 'tumbling over' can't happen as Earth's Rotational  'Moment of Inertia' (pointing) to a fixed point in space is too 'inertial' to tumble; there is no agent that could accomplish it - but in this vast emptiness I don't perceive anything nearby (quarter of Light Year distant) to do it.
Magnetic flipping is possible over several million years; dynamics in the core is poorly understood as of now. No indication to this effect was noticed ever since 'man' evolved or even before it.

Nothing.  The Earth's polarity flips and varies all the time and we're all still alive.

Your compass wouldn't point North, anymore.

It happens because of fluid dynamics in the molten interior of the Earth.  It has happened many times in the history of the Earth.  I don't know the details of why.  To find out why, you should ask a physicist familiar with magnetism and fluid dynamics.
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Magnetic compasses would point the other way. So what?

Earth's magnetic field reverse its polarity from time to time. The last reversal was 750,000 years ago. This is well understood by all but the tinfoil hat brigade. There is evidence that we are in the early stages of a reversal, but there will be no gross effects for several thousand years.