Both of those ("in network" and "out of network") can accept your insurance, and your insurance will pay a portion of the costs.

The main difference is that "in network" has agreed on what the cost of a procedure will be, and exactly how much insurance will pay for (taking into account deductibles and co-pays, and what insurance covers - for instance 80%). There is a fixed total cost.

Out of network means the doctor has not agreed to a price with you and your insurance company. The doctor can charge you much more. So you insurance will pay a certain amount but the doctor will bill you directly for the remaining costs.

For instance, you need surgery and it costs $5,000 according to your insurance company. An in-network place will charge you $5000 (and for instance, you might bay $1000 deductible plus 20% of remainder for a total of $1800).

However, go to an out-of-network place, and they might charge you $10,000. You insurance will kick the same $3200 dollars (maybe, they could just say it is flat out not covered), so you have to pay the remaining $6800.

Keep in mind you are always still paying your premiums for health insurance, but if you don't follow their rules and see their in network people, you typically get screwed. And this is a small example (like getting a wart removed). There can be medical bills in the 10s of thousands or hundreds of thousands dollars easily.

Note: this only occurs in the USA.