What does express delivery in terms of courier mean? Does it mean 'Express' same as in Law I.e. Verbal/written or it means 'Overnight' ?
It definitely does not mean "written". Nor does it necessarily mean "overnight". Express simply means "fast". From Merriam-Webster:
3 a : traveling at high speed; specifically : traveling with few or no stops along the way
b : designed or intended to be used for fast movement or travel
c (1) : delivered faster than usual (2) British : designated to be delivered without delay by special messenger.
Express delivery in general simply means faster than usual delivery. Specific couriers might have particular services that go by this title. For example, until recently USPS had a service called Express mail that guaranteed overnight delivery under certain conditions. The service is now called "Priority Mail Express". But there is no general sense of "overnight" attached to express; you have to ask the particular courier what their express delivery guarantees.