Your insurance will most likely follow you if you get a rental car, but at the limits of your original policy (meaning don't rent a Maserati if you drive a Kia.) If you haven't purchased coverage through the rental car company, then you'll need to file a claim with your insurance provider. Since it's your policy that matters in this case, you will still be able to cover the other driver's car and injuries if you have liability. However, that won't cover the rental car itself. If you bought insurance from the rental company, then your claim for a rental car may be split between your insurance company and the policy you purchased. Some policies will cover the full damages, and some will serve as the primary coverage before your insurance company chips in the extra if you go over the limits.
Limits refer to how much the insurance policy (either your own or the rental cars insurance policy) are willing to pay for certain things (e.g., $10,000 for any property damage and $20,000 for physical injuries.) The deductible refers to how much you need to pay before the insurance company gives you anything. So, if you damage a rental car and the repair cost is $240 and you have a $500 deductible, you will be filing a claim and documenting that you paid $240 to fix it, but you will not receive any type of financial consideration. This is one of the least fun math problems ever.
Does your credit card company or insurance company provide rental car coverage? Possibly.
The major ones (e.g., Mastercard and Visa) might cover towing, theft or damage to the actual rental car, but not property damage or any kind of injury (including another person's car.) In other words, it's a nice safety net but you may not want to rely on it 100%.