if i understand your question correctly, then your "play ins" would be the first 10 games you play when starting ranked, these will determine what rank you start at (bronze 2, or plat 1 for example) then whenever you gain enough LP to go into the next rank (bronze2 -> bronze1) you will play a promo series, this is a best out of 3 you need to win 2....
to move from bronze --> silver you need to do a slightly diff series, the MMR makes it more difficult im told to make sure you "deserve" to move to the next lv.... this info and much more is prob available online fyi
infact i just did a quick search here ya go: http://leagueoflegends.wikia.com/wiki/League_System