Quidem took me forever to figure out, and none of the eighteenth- and nineteenth-century texts and grammars I went to gave me any help at all, because they all said things like, "it means forsooth!" and I was like, yeah, great, thanks, dude.
There seem to be three main contexts in which quidem is encountered. The most frequent one sets up an implication of contrast:
Mārcō quidem magistrī verberātus valde placébat.
Marcus really liked the teacher's beating. (With the implication that other kids didn't.)
Meā quidem sententiā est Mārcus īnsānus.
In my opinion, Marcus is insane. (With the implication that other people may or may not be smart enough to be of the same mind.)
The second context in which you see quidem is in reassuring answers.
Tibi epistulam mīsī Latīnē tam pravē ut verisimiliter nōn scripta Latīnē esse dīcī nequeat.
Epistulam lepidam accēpī, quæ Latīnē quidem scrīpta est.
I sent you a letter in such bad Latin that it probably can't be said to have been written in Latin.
I received your charming letter, which was written in Latin.
The third context for quidem is the last one given by latinlexicon.org, which is pretty straightforward.
Tam gustōsa erat cēna ut nē Mārcus quidem Aurēlius abstinere posset.
The dinner was so delicious that not even Marcus Aurelius could abstain [from it].
I'm sure there are other meanings of and contexts for quidem, and perhaps others can add them in answers, but those seem to be the ones I run into most often.