What tyres do I need for my car?

Well size is everything...as far as your starting point goes...

Check the side wall to identify the correct size of tyres for your car as on the image below. 

A tyre at the wrong size will make your vehicle unstable; there are exception with cars engineered for different front axle and rear axle pairing of tyres but again in this case the pair on the axle needs to be the same size.

What tyre brand should I choose?

In terms of optimum performance the short answer would be choose the original tyres that were fitted to your car. Such tyres have been selected by the car manufacturer to optimize safety and performance. Moreover, some premium car manufacturers develop tyres exclusively for their vehicles alongside tyre manufacturers. These tyres are coded to signify their original equipment status.

Other key considerations 

Price - Budget or Premium

If you generally undertake shorter distances, and cover lower mileage then budget will suffice and conversely premium tyres should be your preference if you rack up the mileage and take long journeys as they will last longer.

Budget tyres are also less likely to perform as well than premium tyres in certain conditions, such as on wet roads giving less effective stopping distances.

Finally - DO NOT BE SWAYED BY PART WORN TYRES. The wear could be unknown and well hidden as could the age leading to dramatically increases chances of rapid deterioration and failure.

Seasonality

Summer tyres, winter tyres or all-season tyres – how do you know which to choose? The type of rubber compound will determines whether a tyre is manufactured specifically for summer or winter use – meaning there's often increased grip and better handling for the different seasons. 

Your Last consideration - Avoid mixing and matching - especially across the same axle!!!

Having different types or ages of tyres on the same axle of your car will lead to imbalances in rolling resistance and grip, affecting how it steers, grips and stops.

For example, a new tyre on your front right  paired with an older tyre (and a different tread pattern) on your front left wheel will lead to uneven grip and rolling resistance.

Avoid mixing winter or on/off road tyres with normal road tyres as they are designed to perform differently in those different road conditions.

AND don't ever mix cross-ply and radial-ply tyres. Their construction is fundamentally different and your vehicle will not have a completely unbalanced stress tolerance - accordingly it is strictly illegal.