"но" is "but" Они были там, но он их не видел — They were there, but he didn't see them. "А" has a few meanings. 1) a = while Родители ушли, а дети остались дома — The parents went out while the children remained at home. 2) а = and Это твоя книга, а это - моя. — This is your book and this one is mine. 3) а = but Это не он, а ты испортил вечер ­— It was not him, but you, who spoiled the party. 4) а = yet Она подтвердила это, а он сомневается — She affirms it, yet he doubts it "Но" normally opposes something to something. "А" more like switches subjects, than opposing them.

In short "но" = "but", "a" = "and"