My son just entered the Terrible Twos. He started saying "no" and throwing toys when he doesn't get his way. My first was an angel until he turned four so not quite sure how to deal with it the right way.
Kristina's Response to Terrible Twos:
You were fortunate not to experience the dreaded "terrible twos" with your first child! But don't worry, your younger child's current state is not a result of some parenting mistake or defect. The terrible twos may be a cliche, but it is a very real stage that many (but not all) children go through. It may be helpful to remember that this is simply one more developmental stage in their process of growth into a self-aware, resilient, fully-functional adult human. This too shall pass.
At this stage in your son's maturation, he is becoming aware of you and others, and their social role in making decisions for him. He is also starting to become aware of himself and his own ability to choose. His job during this stage is to travel down each road, see where it goes, and test the boundaries along the way. Once he has determined that a certain road is untravellable, he will have no more reason to travel down it and will focus on a different route.
Now that he is testing out making choices for himself, your main goal during this time is to calmly show him which roads are safe and productive for him to choose to travel down. You of course want him to choose roads like "obedience to parents" and "kindness to others" so that he is safe and can learn to function with friends, family and society.
An example of keeping this perspective: Let's say he snatches a toy from his sibling. You know his goal is to get to play with the toy, and his brain is trying to figure out the best/fastest/easiest/most efficient way to do that. He is trying to test what choices are best for him to make. If you make sure that "snatching the toy" is NEVER the best way to get the toy, then you will be showing him which roads he doesn't want to walk down (and which roads he DOES want to walk down) to get the thing that he wants. If he snatches a toy, that simply means he doesn't get to play with that toy that day. He may throw a fit, but tomorrow, he'll remember that snatching toys isn't the best way to get the toy, and so he'll ask for it instead, or wait his turn (or whatever positive avenue you show him how to do).
When he's shouting "No!" and throwing a tantrum, it may take a lot of self restraint, but you can just stay calm, keep him safe and let it work itself out. You may have to literally sit down and wait for the storm to pass. If he's in public or throwing a fit in a place that it will be disruptive, you can take him to the restroom or the car, and let him finish his tantrum. Sometimes he may just need a hug or a distraction to calm him down, but if he is in a full-fledged tantrum like this, you can just let it run its course, and let him see through the process that it doesn't get him what he wants any faster. Eventually (not right away, but over time) the tantrums will stop.
At the age of two, he is probably not really ready for a big set of rules, time outs or much "discipline" as it's traditionally thought of. But by utilizing a calm approach, and keeping your perspective during each situation that your goal is not to punish or discipline, but to help him learn to choose the safe/productive/healthy roads, then you will be headed in the right direction!