In this case même is an emphasizer. It means even.

In France everyone speaks French, even the children.

Même also means even not only same

It has an emphatic role in this case to stress that the children speak it as well!

Example:

Même avec des sacs gonflables perfectionnés…

Even with advanced airbags… You may often see that in cars.

In this example it translates to:

In France, everyone speaks French, even the children.

Or “In France everyone speaks French, including children.”

Possible meanings:
If used before an article: even
If used before a noun: same
If used after a noun: very

We use it as adjective to identify something is identical to an other OR is is being use in an expressive manner to put emphasis on something.

The best trick I can give you the recognize it is when, même, is not used as a comparative, it often being use like that : …même qu’il a dit ça. … lui-même.

It being use almost the same as even in english. For example

Oh!, I can’t beleive it !! even him did that ?

Oh, Je peux y croire!! Même lui a fait ça ?

Or sometimes it will be use like you would say himself

This as been written by Mi4_Slayer himself.

Ceci a été écrit par Mi4_Slayer lui-même.

Even the French is my native language, I am no expert in the very depth of it’s grammar, but I hope this help you to understand more of it use in a sentence.

Also I am from Quebec and we tend to multipurpose more the word même more than our cousins from Europe. But I think it will be enough to get the general idea.