Е is a "ye" or a vowel "raising" the preceding consonant (the middle of the tongue is higher; it is called "palatalisation")
Э roughly corresponds to something like "e" in "red". Maybe with a hint of "a" in "rad". At least, Russian speakers hear "map" as something like "мэп" with a vaaary open "э". Also, we have some difficulty distinguishing between "man" and "men", "bad" and "bed", especially in AmE.
Note that in a number of foreign words "е" actually means "э" and is spelt there due to its beauty alone (тест, менеджер, флешка, плеер, проект). It mainly concerns front consonants, usually T, D and S, sometimes M. As words became more "Russian", the pronunciation shifted to the usual pronunciation of е. However, modern Russian feels rather comfortable having both Е and Э, so there is guarantee the same is going happen to the loanwords of the recent decades.
Most operating systems have a built in option to switch between keyboard languages.
On Windows 7, you can go to Control Panel>Region and Language>Keyboards and Languages>Change keyboards.
Then add whatever languages you want to the list, and then go to "Advanced Key Settings" to assign them to button combinations.
If you are on a Windows Computer: Settings-->Time and Languange-->Language-->Русский. (To change it press Shift+Alt) If on a Android phone:Settings-->Languange-->Русский (To change it press the small button on the keyboard that looks like a globe) I do not know how to do it on other devices. You can look it up.