These words refer to something different, remaining, or additional. They both are placed before the noun.
another means "one other" and can only be used with singular nouns. In other words, you can have another drink, job, girl/ boyfriend, apple. in fact anything where you already have one or more, and want to add one. For example: "I've got two children and I'd like to have another." Or: "I'd better not have another cake -I've had a lot already." It can also mean that something is different: "Having a joke is one thing, playing cruel tricks on people is quite another."
other is more used because it doesn't specify a number, and basically just means "additional" or "different." For example: "They have two other children as well as Bobby" or "Would you like any other service?". And it can be used with singular or plural.
It is also used when something is definite, ie "THE other" ("One hand washes the other: we only have two hands, so you would speak of one and THE other, not one and ANother in this case.)
More Examples:
• There are other jobs you could try.
• Where's the other packet of cereals?
• Is there any other bread?
• Have another cup of tea.