Много vs. многие

The word много has several different meanings. In our discussion today we have in mind the meaning of "many, a lot of"; specifically we will have in mind when it is used with plural nouns, not singular nouns. In this meaning we can call много an "indefinite number." Многие in today's context is also used with plural nouns and means "many (but not all)." At first these can be a big confusing because in the oblique cases their endings are the same:

Nomмногомногие
Accмного*
Genмногихмногих
Preмногихмногих
Datмногиммногим
Insмногимимногими

Много means "a large, indefinite number of." When it occurs in the nominative or accusative case, the noun it modifies must be in the genitive plural. In the oblique cases, the noun must be in the same case.

В Америке живут много иностранцев. A lot of foreigners live in America.
Я знаю очень много иностранцев. I know a lot of foreigners.
Нина знакома со многими иностранцами. Nina knows a lot of foreigners.
Иван помог многим инстранцам. Ivan helped a lot of foreigners.

Многие means "a significant number (but not all)" or "many (but not all)."

Многие его идеи мне нравятся. I like many (but not all) of his ideas.
Многие реки России текут медленно. Many Russian rivers flow slowly.
Я лично знаю многих из вас. I personally know many of you.
Нина помогла многим из них, но не хватало времени всем помочь. Nina helped many of them, but there wasn't time to help them all.
Иван познакомился со многими из новых студентов. Ivan got to know many (but not all) of the new students.