Toybox's 1.0 release goal is to turn generic Android into a development environment capable of compiling Linux From Scratch. A tiny system built from just toybox, linux, a C library, and a C compiler (such as LLVM or gcc 4.2.1+binutils 2.17) should be able to rebuild itself from source code without needing any other packages.
In general, configuring toybox for "defconfig" enables all the commands compete enough to be useful. Configuring "allyesconfig" enables partially implemented commands as well, along with debugging features.
Several toybox commands can do things other vesions can't. For example the toybox "df" isn't confused by initramfs the way other df implementations are. (If initramfs is visible, df shows it like any other mount point.)
The toybox shell (toysh) aims to be a reasonable bash replacement. It implements the "sh" and "toysh" commands, plus the built-in commands "cd" and "exit". This is the largest single sub-project in toybox.
The following additional commands may be built into the shell (but not as separate executables): cd, exit, if, while, for, function, fg, bg, jobs, source, alias, export, set, unset, read, trap, and exec. (Note: not done yet.)
The toybox todo list mentions many potential commands which may be added to this project. (Whether that file is readable by anybody but the project's maintainer is open to debate.) The roadmap wiki in the nav bar has a more human readable version.
The criteria for a toybox 1.0 release is that a system built from just the Linux kernel, toybox, C library (such as uClibc), and a compiler (such as tinycc) can rebuild itself from source code.
Most commands are implemented according to The Single Unix Specification version 4 where applicable. This does not mean that toybox is implementing every SUSv4 utility: some such as SCCS and ed are obsolete, while others such as c99 are outside the scope of the project. Toybox also isn't implementing full internationalization support: it should be 8-bit clean and handle UTF-8, but otherwise we leave this to X11 and higher layers. And some things (like $CDPATH support in "cd") await a good explanation of why to bother with them. (The standard provides an important frame of reference, but is not infallable set of commandments to be blindly obeyed.)
The other major sources of commands are the Linux man pages, and testing the behavior of existing commands (although not generally looking at their source code), including the commands in Android's toolbox. SUSv4 does not include many basic commands such as "mount", "init", and "mke2fs", which are kind of nice to have.
The 2015 toybox talk starts with links to three previous talks on the history and motivation of the project: "Why Toybox", "Why Public Domain", and "Why did I do Aboriginal LInux (which led me here)?".
It's carefully stacked soda cans. Specifically, it's a bunch of the original "Coke Zero" and "Pepsi One" cans, circa 2006, stacked to spell out the binary values of the ascii string "Toybox", with null terminator at the bottom. (The big picture's on it's side because the camera was held sideways to get a better shot.)
No, it's not photoshopped, I actually had these cans until a coworker who Totally Did Not Get It tm threw them out one day after I'd gone home, thinking they were recycling. (I still have two of each kind, but Pepsi One seems discontinued and Coke Zero switched its can color from black to grey, presumably in celebration. It was fun while it lasted...)