There are two types of insulin production in beta cells: baseline and triggered by elevated blood glucose (and other stimulants) levels. This on demand release is quite quick (approx 10 minutes) because it involves storage release as well as initiating new production.
Destruction or dysfunction of beta cells (combined with other factors) is the cause of diabetes mellitus type 1 and diabetes mellitus type 2 respectively.
Apart from insulin, beta cells release into bloodstream (in equimolar quantities) so called C-peptide that can be regarded as a by-product of insulin production. Measuring C-peptide can give an idea of viable beta cell mass.
A rare type of cancer derived from beta cells is called insulinoma. It results in recurrent and prolonged attacks of hypoglycemia.
See also: Diabetes dictionary