Different airdrop and giveaway have their own requirement.  Some of the airdrop need own wallet address. In this case you need to download their wallet and provide your wallet address. In other case they ask your Erc20 wallet address, you need to provide erc20 compatible wallet address. You can try use MyEtherWalle address in this situation

To receive your new airdrop tokens, you should ADD them to your wallet.
Step 1:  find Add TOKEN/ Add Custom Token button.
Step 2: fill contract information, Token Symbol and Decimals
after save the Token detail, then tokens will be added to the list of tokens.
You can check them in your balance now.