Our company healthcare is different to the copay-type ones listed above. It's a high deductible model, where we have an excess that we have to pay before the insurer coughs up. Ours is $3k per person or $8 per family, whichever limit we hit first. Preventative stuff is free, so we get well person check ups with blood work once a year, all vaccinations, screening such as Pap smears, etc.
So this means, for example, that when we took my daughter to the docs for an ear infection, we got a bill a month or two later - cost for the consultation and antibiotic was a little over $100.
We also have dental, which includes all the basic stuff like check ups, fillings and cleanings, and we make a 20% contribution to more complicated stuff - I paid $200 towards a crown recently, with the remaining $800 or so going to the insurer. My son is currently in braces, and our cost towards this is about $2000, spread over 2 years.
If you have a high deductible insurance plan, you can also have a Health Savings Account (HSA). You set an amount to be paid into this each month from your pre-tax wages, like you would with a pension in the UK, and can then use it for any medical related expenses. So we use it to pay for the sort of things outlined above, along with new glasses and contact lenses for me, that sort of thing (it can't be used for over the counter stuff like paracetamol, just things for which you've seen a 'doctor' of some sort). It's really easy - it's just a bank account with a debit card, so you pay electronically like normal, and then keep the receipt in case the IRS ever ask you to justify your expenditure to make sure it's genuine and you didn't use it for that month's groceries.
Because HSAs are funded pre-tax, it makes these costs about a third cheaper in real terms. You can put in up to... um... about $6k a year, depending on single or family, and any money unspent rolls over to the following years. We're planning to fund ours up to the level of the deductible on our insurance, then stop.
More and more companies are going down the high deductible/HSA route; my husband's, a large multinational, has just introduced this system in 2012. The idea is the govt/company wants you to (a) stop bothering the doctor for trivial stuff - if you only have to pay a $20 copay, you can pop in weekly for no real outlay, AND stay and chat for ages. Now, you're paying for the whole consult, so you'll only go when you need to, and be out as quick as poss without a bunch of reassuring but largely unnecessary placebo tests. And (b), they want you to have built up a HSA cushion to pay for some healthcare needs when you're old.
There are some tweaks to the model, where you pay a little more or less in premiums depending on whether you agree to name a doctor, or use their in-network doctor they allocate you (we did this, and have been delighted with the extensive, modern clinic we got; appointments are easy to get, the doctors are great and the clinic is like a mini-hospital with out of hours emergency care and the ability to run tests up to MRI level!).
As far as I'm aware, if your company offers this model, then that's what you'll have - certainly in our case it was a choice of take the company insurance, or arrange our own; there was no ability to negotiate or amend beyond 'do you want to save money by using our network doctors?' and 'do you want to add family dental for around $10-15 a month?' (your answer to that one should be 'oh yes!'). It's a little tiresome if you're healthy with no chronic conditions, because barring an accident you'll just be paying out the premiums and never seeing any benefit from run of the mill illnesses. Conversely, those premiums are of course cheaper, as the insurer knows a lot of people won't be claiming anything back, and are just insured for 'the big one'.