There must be a dozen ways to store user data in iOS. Here are several:
Property lists: An easy way to store a graph of common data storage objects and containers. This is a good place to start if you're just learning the iOS ropes.
NSKeyedArchiver and NSKeyedUnarchiver: Provides an easy way to serialize and deserialize your objects to/from a chunk of data, which you can then write/read using NSData's methods.
NSFileHandle: Read and write data in whatever format you like using a nice Objective-C API. More generally, you should read up on the iOS file system.
UIDocument: A full-featured starting point for managing user data, including syncing with iCloud.
Keychain: Not a general purpose data storage mechanism, but if you're storing sensitive items like passwords, credit card numbers, etc., you should use the keychain API.
POSIX file API: Good old C file handles with the read and write functions you learned in college, if you went to college before Java was a thing.
SQLite: According to the web site: "SQLite is the most widely deployed SQL database engine in the world."
Core Data: A powerful (but also somewhat complex object graph manager. This is a good choice if you have many different pieces of related data to store.
What would be the best/most secure way to store the users information
so that when they close the app it is still in the app. The data
populates a UITableView and is a NSMutableArray.
Best is subjective -- you'll need to consider your needs and look at the various options. For many people, though, best means least painful or easiest to learn. As mentioned above, property lists may be the way to go in that case. If your array contains simple data (strings, data, dates, numbers) in standard containers (arrays or dictionaries), your file I/O can be as simple as something like this:
// writing
[myArray writeToFile:somePath atomically:YES];
// reading
myArray = [[NSArray arrayWithContentsOfFile:somePath] mutableCopy];