Suppose someone has a medical insurance policy from an insurer. He visits hospital and bill is of $25. He claims it from the insurance company. Company pays but they incur $400 on the formalities (paperwork for example) to grant $25. Hence it costs the company $425. From where these $400 be recovered? One way to recover such expenses is to raise the policy premium, but that would mean raising the expenses for the customer (or losing potential customers). An alternative for insurance company is to tell the customer upfront at the time of policy purchase that small expenses up to certain amount will not be reimbursed and premium can be reduced slightly to compensate for this. This is what Deductible is. Deductible is that certain amount. Deductible is also deducted from the total expenses while issuing the claimed amount in case of larger than deductible expenses. For example if policy holder spends $2000 in hospital and the deductible amount is $200 in his policy terms then he can claim $1800.
There is another form of Deductible which is not in monetary terms. It is in terms of time. If a person is disabled due to an accident or some other incident disability insurance does not pay for the entire duration of disability. They usually have a waiting period in terms of number of days for which person must be disabled before the insurance will pay for the remaining time of disability. This waiting period is called as elimination period. So if the person is disabled for 60 days and the elimination period is 10 days then disability insurance will pay the benefits for 50 days only.
Coinsurance is another concept similar to Deductibles to keep the premium low. Coinsurance is common in medical insurance policies. It defines a limit above the Deductible amount until which insured would pay a percentage of the medical bill and insurance company will pay the rest of it. For example if deductible is $200 and coinsurance up to $5000 above the $200 is 10%/90% that means insured will pay 10% of the bill up to $5000 and insurance company will pay 90% of it. So if total bill is $4200 then insured will firstly pay the deductible which is $200. From the remaining $4000 insured will pay $400 (10%) and remaining $3600 will be paid by the insurance company.