The latest version of Android is Lollipop, and Google has made a few changes to the way you access Wi-Fi settings. Signing into Wi-Fi networks on Android varies from phone to phone (and tablet) as manufacturers tweak and change the interface. We're using a Moto G 4G here, but the process is similar on most Android devices. 

How to sign into a Wi-Fi network on Android

In Lollipop, swiping down from the top of the screen brings up a redesigned notification area. You have to swipe down a second time to bring up the toggle controls for Wi-Fi, Bluetooth and more:
How to sign into a Wi-Fi network on Android
However, tapping on the Wi-Fi icon here will enable or disable the wireless radio, so to connect to a network you need to tap the cog icon at the top of the screen.
Now, tap Wi-Fi under Wireless & networks to show a list of available devices and hotspots.
How to sign into a Wi-Fi network on Android
Tap on the network you want to join, and enter the password.
How to sign into a Wi-Fi network on Android
When connected you can tap on the network name to see extra details such as the connection quality, speed, frequency and more.
How to sign into a Wi-Fi network on Android
If the network is unsecured - i.e. no password is needed to connect to the Wi-Fi hotspot - but requires you to log in (such as free public Wi-Fi in an airport or paid-for internet in a hotel), you might see a notification saying "Sign into Wi-Fi network" and the name of the network below it.
Tap the notification to open up your web browser. A page should open with the access point's instructions for logging in. You'll have to fill out any details required, possibly choose a payment option and enter your payment details.
It's always worth remembering not to use any banking websites or do any online shopping where you have to enter your username, password, credit or debit card details over an unsecure network.