If you wish to drive a car that you are not the registered keeper of, you are likely to get turned down by most insurance companies. However there are several ways of getting insured so that you can do so legally.
You may well have already made an application for car insurance for a car that belongs to someone else and had it refused. This is because a lot of insurance companies do not want to know about this type of business because they see it as being more risky than covering a car for an owner driver.
Some insurers will provide cover, but only if the registered keeper is a spouse, partner, other close relative employer, or a lease company.
There are others that will still provide cover even if these criteria are not met but their premiums will usually be more than they would charge under normal circumstances, in order to compensate themselves for the extra perceived risk.
If you only want to insure yourself for a short period you can take out short-term car insurance and all things being equal you can insure yourself for a car whether you own it or not. Short-term car insurance is usually limited to a maximum of 28 days at a time, although it may be possible to buy new policies as the old ones expire. This is a more expensive way of getting insurance, but it could be cost-effective if you only need the cover for a few days or weeks. You can get quotes for this at this short term car insurance site.
Do you already have a car insurance policy which covers a car that you do own? If so, you may well have cover for any vehicle that doesn't belong to you. Make sure you check your policy carefully to ensure that this is the case.
This will almost certainly cover you for third party only risks however. If you are involved in an accident your insurer won't pay for any damage to the car you're driving. The car must also be properly insured by someone else, otherwise your insurer will suspect that you are possibly trying to stretch your policy to cover more than one car.
Alternatively, the owner's insurance company may be prepared to put you on the policy as an additional driver. This can be a very economical way of solving the problem, and some insurers make no charge for this whatsoever.
Either way, it is essential that you inform the insurance company that you are not the registered keeper of the car. If you fail to do so they are perfectly within their rights to cancel the policy and refuse to pay out on any claims that may be made.