Get the other toy. Go the other way. Pick the other snack. When your kid looks at you blankly, you'll wonder if you've got a little person or an alien in your home. Teaching tots simple commands like "no" can be hard enough. Try to teach an abstract qualifier like "other" and you may as well be speaking Chinese. But, at ages 3 and 4, kids' understanding of language and their vocabulary grow by leaps and bounds.
No single teaching method expresses the meanings of words to preschoolers more effectively than talking to them. If you want them to understand the word "other," use it as consistently as you can in random conversations with your child. Use small sentences where the meaning of the word is clear and the other words and syntax don't hamper its definition. You can intermingle the word in your everyday life. Saying to your child, for instance, "Remember the other day when we..." indicates not this day, but a different day -- a specific different day. Or you could ask her questions. "Who was the other child you were talking about yesterday?" She has to process what your question meant to answer it, and when she starts answering correctly without pause, she's beginning to grasp the concept.
You can easily show your child what "other" means by taking it from the abstract realm and turning it into something concrete. Commands work well for this. "Get me the other square. No, not this square. The other one." After he has retrieved the square, try another shape. "Get me the other shape on the table. No, not the squares. The other shape. The triangle." These sentences show him the meaning of the word in physical and definite terms.
Nothing engages a kid like a game. Try branching out from simple commands and make it into a game for your preschooler. Intersperse "other" into your directions when directing her to find objects or toys. You can go on a scavenger hunt together. Remember to continue to talk to her; explain aloud what you are doing and what the next step will be. "Now we are going to look for the other pair of glasses. We found Daddy's. Where are Mommy's? We have to find the other pair." This will help her use the context clues to put the word in an easily definable state.
When you think your child is ready to take the lead and has a good understanding of the term, you can start asking him questions to test his knowledge. If he comes home from preschool and uses the word "other," ask him what he means. An example would be: "Mom, the other kids said I was smart!" Then you ask, "What other kids?" When he lists the children to whom he was referring, mention a child that he didn't. "So you mean, Alex, Kim and Brian said that, but not Ian?" Then he'll most likely tell you why Ian didn't compliment him, but if he doesn't, ask him. He'll think of the reason -- perhaps Ian was not there at the time -- and his mind will have successfully used other as a category without him even knowing it.