As has been mentioned, every car and every tire will be different.
In the case of F1, every tire is the same, so the amount of pressure with vary by team based on chassis design, track conditions and driver profile. Every single car will be setup differently to suit each drivers needs. If a driver wants more oversteer, the front pressures may be reduced slightly or the rear pressures increase slightly etc... On certain cars, they may have to run low(er) pressures consistently compared to other teams due to design limitations.
Here is a BASIC!! rundown of how tire pressure can affect traction. Keep in mind, this is BASIC theory only
If the tire pressures are too low for example, the tire will exhibit more sidewall flex. This is detrimental because the flexing of the tire in an extreme amount allows the car to move relative to the contact patch. Think of a flat bicycle tire... you are able to rock the entire bike side to side whist the tire's contact patch remains in the same position. This causes a degree of unpredictability and the car will feel like it is wobbling down the track.
Keep in mind, this is always going to happen, regardless of tire pressure, but the less it happens, the better/more predictable the handling of the car will be.
Another problem with running low tire pressures is that the contact patch tends to actually become two small contact patches at the very edge of the tires. You'll notice on a regular car tire that has been under-inflated for long periods of time, that the center of the tread is relatively un-worn while the outside edges are usually quite badly worn. Making two contact patches work together can be incredibly hard, if not impossible.
Over inflating a tire causes exactly the opposite of both of these problems. Sidewall flex, though still present, becomes too little! This is bad because more and more of the effect of a bump will be transmitted to the suspension. This can cause the car to bounce, the tires to hop over bumps instead of absorbing them, or generally just drive unpredictably over rough surfaces.
Also, the more inflated a tire becomes, the less of a contact patch you will get in return. Again, using a car tire as an example, one that has been over-inflated for a long period of time will show you that the center of the tire is very badly worn while the outside edges are hardly worn at all.
So, where does that leave you? The optimal goal is to use as much of the tire as possible at all times while cornering. Using a large portion of the tire keeps wear consistent and temps lower. That is where camber and suspension geometry come into play to work along side the tires, but thats another thread all together...
Obviously, there are exceptions to the rule, again, depending on design and drivers etc...