the one in the second picture totally looks like a nano, and I have used them a lot. They're basically a stepped down version of the basic Arduino UNO, almost fully compatible on software level, and with very minor hardware differences (of course some tweaking is necessary if you wanto to use a regular arduino shield with it. Chinese clones are as cheap as 3/4 $. They're very good and I'd recommend them for projects where space is important. I also have managed to connect them to the internet with the very cheap ENCJ chips. With less than 10 $ you can have a full working "internet of the things" gateway.
I built a remote thermostat with that, connected to the web and with a temperature and humidity internet log and a web interface for setup.
The one in the first picture is instead an AtTiny85. It's a chip from Atmel (the same that manufactures the Arduino chip). It's only partially compatible with Arduino mainly due to very strict limitations on the hardware side. It's very tiny as the same suggests and you can do some good things with it where the space is very low (the one you posted is a board with the ATTiny and some other non necessary component on board, you could just strip it down to the main chip, but the price is just a little less than the Arduino Nano (so not worth the hassle in my opinion); also it needs a USB hardware interface for programming, while the Nano is (almost) plug and play in every PC. I never used one of these.
If you need help with them feel free to ask.