The "weak form" of "a" is an unstressed schwa /ə/, the strong form is stressed and pronunced as /ei/ (the same as the name of the letter) The strong form is used when the indefinite pronoun "a" is used in a contrastive way.

> The UK will join *a* customs union, but not *the* customs union.

(pronunciation could be /ei/ and /ði:/. This is not mandatory and the stress can be made shown in other ways. The contrast here is between the indefinite and definite)

The weak form of an is unstressed /ən/ and the strong form is stressed /æn/ (the same sound as in "man") Again it is stressed to show a contrast.

> I said I wanted *an* eclair. I can't eat 5!

In this case the contrast is between 1 and 5.

Not sure what you mean by British and American. The dictionary site you link gives American strong and weak forms as being the same as British strong and weak forms. (Phonetically the same, but the actual realisation of those phonemes is different)