If you mean the names as we learn them in school, then they're very rarely written down (so I'm effectively inventing these spellings), but they would (in British English) be:

• Ay /eɪ/
• Bee /biː/
• See /siː/
• Dee /diː/
• Ee /iː/
• Eff /ɛf/
• Gee /dʒiː/
• Aitch /eɪtʃ/ — sometimes pre-aspirated as Haitch /heɪtʃ/, though this is considered incorrect in RP
• Eye /aɪ/
• Jay /dʒeɪ/
• Kay /keɪ/
• Ell /ɛl/
• Em /ɛm/
• En /ɛn/
• Oh /oʊ/ or /əʊ/
• Pee /piː/
• Queue /kj(ə)uː/
• Ar /ɑː/ — with varying degrees of rhotacism; I grew up in the West Country, so say /ɑʵː/
• Ess /ɛs/
• Tee /tiː/
• You /j(ə)uː/
• Vee /viː/
• Double-you /ˈdəb(ʉ)l.j(ə)uː/
• Ex /ɛks/
• Why /waɪ/
• Zed /zɛd/

I get what you mean by asking about a is for apple; whilst I'm sure I was taught something like that when first learning to read, it has been so infrequently used since then (I'm now 35) that I honestly couldn't tell you what they were. The letter names above, though, are how I would spell out my name to someone unfamiliar with it, for example.

The Nato Phonetic Alphabet (Alpha, Bravo, Charlie, Delta and so on) would be familiar to some, but not to all — my mother would probably think me pretentious for using it. But that is what our police would use to provide clear, unambiguous details of a car number plate over the radio, for example.