Teaching responsibility to any child is an on-going process. "Responsibility" simply means that one is depended upon to perform a chore or a job or a duty, and that one is held accountable for the outcome.
Children learn by seeing, doing, hearing, and imitating, as well as by more formal methods. This begins from birth, when a child is learning about himself, and progresses through the toddler years when a child begins to realize the place of self in the larger context of his family.
Even babies who have not yet begun to walk can observe a parent placing toys in a basket after playtime, and often will imitate what they see by "helping"; of course, this "helping" can involve removing the toys as well!
Once a toddler has developed confidence in walking and frequently begins to carry objects while doing so, he can be taught to do many things. The earlier a parent begins this process, the easier it will be and the more habit-forming it will become. Things to keep in mind are that toddlers will need reminding when it's time to do a chore and supervision during the work itself; in fact, kids of all ages will require these two things and not until they've moved out on their own will they actually (hopefully) take full responsibility for daily chores and jobs. The key is that toddlers really, truly love to help, and if a parent begins when the child is young, the chores become habit and much more hassle-free throughout the years.
The list of what small children can accomplish is long: placing spoons and napkins on the table, picking up trash from the floor, putting their shoes in the closet, brushing the dog. They may not do any of these things well, from an adult's standpoint, but they can do them. The more frequently this occurs, the better they will become at various tasks.
Three-year-olds are quite capable of being taught to fold their own clothes and make their own beds. They can clear most items from the dinner table. They can pick up their own toys, and they are able to help with less-dangerous kitchen and cooking duties and even wash dishes. They can rake leaves, or help paint a wall.
Children should be allowed to have fun and be "kids", yes, but they should also be expected to contribute to the household in which they live. The bonus is that most of the time, most children think that these things ARE fun!
How does this relate to responsibility? The answer is twofold: accountability and praise. Accountability stems from allowing the toddler to actually do the job, with minimal assistance, and to be allowed to fail, if necessary. For example, if the family sits down to dinner and there are no napkins, the toddler should be requested to bring them - this is his job, and the parent has already "reminded" him. Of course it doesn't hurt or inhibit the toddler from learning if someone else goes to get the napkins, but that should be a rare occurrence. If your child's job is to do something, let him do it. This is the crux of responsibility.
Praise in itself can be very controversial. Today's children are all too frequently commended for little, silly things which they should already be doing. One might hear how little Johnny was "awarded" a ribbon for merely putting on his socks; or that little Susie was given a piece of candy for hanging up her jacket. Yes, children need to be recognized for their accomplishments, but the giving of awards and constant gifts will only backfire on the parents. Children need to be taught the value of an intrinsic reward, the satisfaction of a job well done.
You should, by all means, praise your toddler for completing a chore, whether or not it's done to perfection. Tell little Johnny, "Good job!" and be specific about how it was a "good job". But don't ignore the negative aspects entirely. If Johnny picks up all the dirty clothes in his room and puts them in the basket, but misses a pair of socks and a shirt, tell him, "Good job, Johnny, but you missed those socks and that shirt, so pick them up too and you're all done!" Let him finish the job, and move on to the next activity.
If you don't let him finish, he will think it's not that important; no, it's not imperative that the rest of the items join the others right then and there, but the job itself is not finished until they do so. Teach the child from the beginning that an incomplete chore is not really a help if the parent must finish.
At the same time, don't go overboard in praising Johnny. The completion of the laundry-gathering chore does not require effusive praise and reward. These are Johnny's clothes, and he should learn to take care of them as he learns to care for all his possessions. If the parent says, "WOW! Johnny, you did such a GREAT job picking up your clothes, you are AMAZING, thank you SO much!" and so forth, little Johnny will soon be expecting vociferous praise for every tiny thing he manages to accomplish, such as getting up in the morning or pushing in his chair after lunch.
Teaching responsibility to toddlers starts early and must be consistent. Honest, simple praise coupled with reminders and supervision are necessary to ensure success for the child, but overblown rewards can certainly be too much of a good thing and will backfire as the child grows and matures. There is nothing wrong about a child who realizes that he is a part of the family and therefore must contribute. The sooner one begins, the more habit is developed; this will benefit the entire family.