Modern tyres are marvels of chemical and physical engineering. They are more resistant to punctures, have better grip, smoother ride and stronger sidewalls than anything we could produce 20 years ago. But it is still quite possible to overwhelm them when you demand too much. Misreading the road conditions, poor judgement, inappropriate maneuvers or use of speed can all cause issues. It all boils down to one simple rule though, and if you live by this, your tyres will thank you for it. Plus, you'll have a lot more fun in corners - even turning into your street off the main road can be a fun couple of seconds. The rule is this : don't brake and corner at the same time. Get all your braking done in a straight line, then turn, and start to accelerate when about half way through the turn as you begin to straighten the wheels again.
By doing this, you afford your tyres the best chance of gripping for both situations. Braking in a straight line gives you predictable, safe behaviour. The initial half of the turn (without braking) allows your tyres a good opportunity to grip as you turn the wheel and the car begins to come around. As you begin to straighten the wheel, start accelerating again - the tyre is dealing with less cornering force and can provide more front-to-rear grip again.
The explanation for this can be complicated but I'll try to simplify it. Tyre treads and construction methods, coupled with rubber compounds, mean that modern tyres have very good grip in two axes - forwards-backwards (linear), and side-to-side (transverse). The linear grip is used for most driving - accelerating and braking. The transverse grip is used when cornering to help stop the tyre skidding sideways in the turn. If you attempt to brake and turn, or accelerate and turn at the same time, the thrust vector on the tyre's contact patch is moved into an area between the linear and transverse areas of performance, giving you diminished performance in both. This results in 'understeer' whereby your car's wheels are turned, but your car (and you) are not going in the direction you want - you're going wide in the corner.
Understeer is most often caused by two things - a slipperier than normal road surface (ice, snow, rain, oil) or a driver who goes into a corner too fast for the conditions and thus keeps trying to brake whilst cornering.