For a bitcoin wallet you can choose between several types of wallets:
1) Desktop wallet
1.1) Full node (bitcoin core [1])
1.2) Lightweight wallet (e.g. electrum [2])
2) Mobile wallet (e.g. Mycelium)
3) Hardware wallet (e.g. ledger, trezor)
4) Paper wallet
Paper- and Hardware wallets are the safest options, since your private keys won't ever leave a safe "environment".
A paperwallet is basically for free, but can be quite demanding when using the first time. Where a hardware wallet is easy to use, but costs a few bucks (50$+).
Mobile and desktop wallets are roughly equally safe (You shouldn't store big amounts on them).
A fullnode wallet (BTC Core) downloads and processes the whole blockchain (150GB+).
A lightweight wallet does not need to download the whole blockchain. It uses an online service to communicate with a fullnode to get the current balance of all of your addresses.
[1]
https://bitcoin.org/en/download[2]
https://electrum.org/#home