In simple term, Mist browser is user interface for Geth client. Geth is one the Ethereum client that is used to connect to Ethereum networks and it provide command line utility to do various functions. Mist browser provide user interface to perform similar functions that we perform using Geth client command line utility.
To download the mist browser, go to below link.
Run the exe file and install the mist browser. Once you open the mist browser, it start connecting to peers/nodes and start downloading the blockchain data. Click on launch application link to launch the mist browser.
Once you open it, you can go to "Develop" menu bar and check the "Ethereum Node" option, that shows which version of Geth is being used.
You can go to "Network" option under "Develop" menu bar option to choose to which blockchain network you want to connect.
You see one more option "Sync mode" that show the mode that you would like to choose to connect with blockchain network.
You can also create accounts and develop smart contracts under wallet option as shown below. You can go over the video for more details.