The way ranked works is there are 5 ranks to a tier. Lets take Bronze for example.
Say you were placed in bronze 5. You will start usually with 0lp and work your way up. Once you hit promo's if you've been winning a lot you can skip a division because the system thinks its placed you in the wrong division or tier. In this sense you can jump from Bronze 5 to Bronze 3 from 1 promo. This doesn't apply if you've been playing a lot of games in 1 area like say plat 3 and finally reach promo's. Those promo's wont skip you to Plat 1 because you've lost a lot of games and the system believes you are climbing at a rate that fits your skill.
You won't demote immediately because Riot has said you have a short buffer period where you can't be demoted, BUT it does mean you can lose mmr. Remember how I mentioned the system figures it placed you wrong and will skip you ahead a rank in your tier? Well if you continue to lose on 0 lp the system will believe it placed you in the right location and will lower the amount of lp you gain per win and increase the amount you lose if you drop below a certain average mmr range for your tier / rank.
Say you're a platinum player but you continue to lose games. The system will lower your mmr and pair you with higher ranking gold players or even silver players that have played very well (generally smurfs or players on hot streaks) Your mmr (matchmaking rating) is at their level even though you have a higher tier (platinum vs their gold) doesn't mean you have a better mmr. Think of it this way, if you win consistently, say 6 out of 10 games every 10 games you play you're about average even 5 wins then you will climb.
The game won't punish you for immediately going into a new rank, just don't be afraid, play nice, and play for fun, cause this game is meant to be fun not a stress hive.