There are two types of wallet apps: local wallets and cloud wallets.
Cloud wallets store your bitcoins keys on the internet. It is convenient, can't be physically "lost", and can be accessed by multiple devices (phone, computer, tablet, whatever). Coinbase has a pretty good bitcoin app that is "cloud-based". You can buy bitcoin and ethereum directly on the device, and it's insured from theft and hacking. However, this app is not private . Another good cloud app I like is blockchain, but it unfortunately does not provide bitcoin/ethereum purchases.
The downfalls of cloud wallets are that:
a) Cloud wallets are not private - the company that manages the cloud wallet will know who you are paying by analyzing the bitcoin addresses
b) Cloud wallets are easier to "hack" if the perpetrator guesses your password
Use this option if you need things to be kept private. Local wallets store your bitcoins on the phone itself, so corporations and nation-states can't meddle with it. The private keys are kept locally on the device's hard drive, so you need to backup the wallet, otherwise, you can lose your bitcoins! Mycellium and Electrum are good wallets to use. They're safe, and relatively easy to use.
Do not use any wallet from a company called "Freewallet". It is a probably a scam
Taken straight from reddit:
"I'd suggest the community install (without sending ether) and give them 1 star rating until they identify and prove to the community who is behind the project and make other important changes to prove trustworthiness. This would be a start. The way it stands right now all signs are pointing to scam. No names visible online or in support emails, they believe holding customer funds is a benefit to their users, playing off Ethereum branding to make it seem like it's an Ethereum proper product. Please be very careful."
The Bitcoin Spot
5152 Lankershim Bl
North Hollywood, CA 91601
Hours: Monday - Friday 9:00AM - 5:00PM