These are more convenient and easier to use than hardware wallets, and they are free to get started with, but we do not recommend that you store thousands of dollars worth of coins on them. Use them for day-to-day use, or for storing small amounts of money, because they make it so easy. But NEVER EVER store large amounts of money, or your entire savings, online or on your mobile. EVER! You have been warned. Why? Because online and mobile wallets can be hacked. Bitcoin cannot be hacked, hardware wallets cannot be hacked, but online and mobile wallets themselves can be hacked, and when they are, you can lose whatever you had stored in them. In addition to hacking, governments can shut down or seize online wallets and their passwords by simply grabbing the wallet provider's database. So, ALWAYS transfer large amounts of coins to a hardware wallet.
Generally, online/mobile wallets are relatively safe; most people live well with them without any incidents. But because they are online, they do stand the risk of a hacking attack. That wold rarely happen successfully, but if and when it ever does, you don't want your big nest egg sitting online at the time. Hardware wallets are offline.
So, the rule of thumb is: online/mobile for convenience with smaller amounts or regular trading activity, hardware for long term storage of large amounts.
There are literally dozens of wallets! A Google search, or an Android/iOS marketplace search, will give you lots of options, some of which are unsafe and untested. Therefore, instead of listing everything available out there, here are some of the more recognised and safest ones:
Here is an example of how a mobile wallet looks like:
There is no "best wallet" for everyone. Each wallet has different features and, depending on your needs and tastes, you will, through testing each one, find the one that feels best for you. Many people use more than one wallet; it can be safer and more convenient.