The most important thing with discipline is praise the good behaviour and ignore the bad behaviour.
I would explain things to an 18 month old baby, they know a hell of a lot more than we give them credit for. My daughter is 21 months and I'm always talking and chatting to her about what's right and wrong. How else will they learn if you don't explain things to them. Also I wouldn't leave disipline until they are 2, because it might be a bit late to suddenly be firm with them. Never underestimate the understanding of a toddler. Just do it a bit more gently at a younger age.
Ignore tantrums, they only do it when they have an audience!! Take the audience away and there's no one to perform for.
Give them consequenses to their actions, ie: give them 2 choices, do you want to walk along like a big girl/boy or would you like mummy to carry you there. If protesting and wont walk. Then they learn there is a consequence to their actions.
Never underestimate the power of not talking, use actions rather than words.
inclusion/exclusion, they hate to be left out of things or things taken away, use that to your advantage and explain at the same time. "If you are going to fight over that toy, I will take it away until you decide to play nicely" give them the choice to decide!
right, I think that's enough waffling, I've just completed a positive parenting course, very useful, would recommend to anyone.