When you receive this pop up information, with the error Google play services has stopped, especially if you are still using the Android Lollipop version, then you need not panic.
Just follow this steps and you will solve your problem 80 percent of the time.
1. Do a Soft reset.
Yea just holding the power button and doing a soft reboot may end up solving the problem, because sometimes too much open apps may cause the RAM to become crowded.
2. Uninstall Google services update
Go to Settings > Apps > Google play services > then uninstall updates. Then Restart.
You can update google play service if you are using an old version. Download the latest version here.
2. Clear cache or and framework app
Depending on your device, Google services may allow for clearing cache.
To clear cache head to Settings > Apps > Google services then tap on the clear cache.
For framework app Repeat the steps from method 2, but this, time do it for the Google Services Framework app instead.
3. Remove and re add your google account
Go to Settings and then Accounts & sync. Select Google from the list and tap on the one you use for the Play Store. .
Tap on your email address it opens a new menu if you tap on the 3 dots at the top right of the screen you should see the option Remove account. Select that and then confirm it.
To re add your account back to account and follow the steps to add it back.
If those fails try this other methods
7. Reset app preference.
To do this Settings and then Apps. On the screen listing all of your apps, tap on the 3 dots at the top right of the screen. In the new menu that appears, tap Reset app preferences. Now restart your phone
8. Do a factory reset.
If everything fails try a factory reset, be sure to back up as all your data stored on phone memory will be wiped off.