A premium is the amount paid each month to have your plan regardless of whether you "use" it or not.
A deductible is the amount you pay for health care before your insurance kicks in. For example, if your deductible is $1,500, you will pay the first $1,500 in medical bills out of your own pocket.
Deductibles are cumulative over the course of a calendar year – each time you pay a small bill, it counts towards your deductible. Once you've reached your deductible, your insurance will begin to pay the majority of medical expenses. Each year on January 1st, your deductible resets and you start all over.
Every time someone in the family pays for medical care, that cost will go toward the family deductible. But it will also go toward the individual deductible of the family member who received care.
If that family member hits their individual deductible, insurance will kick in and begin to share the cost of medical care for them (and them only). Everyone else in the family will still be responsible for paying for care out-of-pocket until they've either met their own individual deductible or the family fulfills the family deductible.
A copayment is a flat-fee you pay for treatment. Copays are a type of coinsurance, only they are a flat-fee rather than a percentage. You usually pay copayments when you check-in at a doctor's office, other medical facility, or get a prescription (unless you're receiving free preventive care services). A copay is often a small amount (like $20 for a typical doctor's visit).
Coinsurance is the percentage of the bill you pay. After you have paid your deductible, your insurance company will cover the majority of your health care costs and you are only responsible for a "coinsurance" portion. For example, suppose your plan has 20% coinsurance, you've already paid your deductible, and you need a $10,000 surgery. You would pay a maximum of $2,000 and your insurance would pay the rest.
Out-of-pocket expenses are the amounts you will pay, in addition to your monthly premium, if you require medical care.
The maximum out-of-pocket amount is the most you'll pay for in-network healthcare, excluding monthly premiums, in one calendar year. Every time you spend money on health care (in the form of deductibles, copayments, and coinsurance), it counts towards your maximum out-of-pocket. Once you reach this amount, your insurance company will pay the full cost of any covered medical service as long as you keep paying your premium.
Let's look at an example:
Suppose you require a $100,000 heart transplant and you bought a plan with a $5,000 deductible, 20% coinsurance, and a $10,000 maximum out-of-pocket. Assume you haven't yet contributed anything toward your deductible. You would only pay $10,000 for the surgery rather than your much higher coinsurance amount. This is because you are protected by your maximum out-of-pocket amount.