あげる is a verb with many uses in Japanese. It’s original meaning is ‘to raise’, but is also used as a polite 丁寧語（ていねいご） way of saying ‘to give’. In this way it can be thought of as similar to ‘to offer up’ in English. As teineigo (polite speech) has the same basic subject and object rules as standard speech, the giver will be marked with
は or
が, while the recipient will be marked with
に. As usual, the ‘object’ that is being given will be marked with
を.
-
トムがタカにプレゼントをあげる。
Tom will give Taka a present.
-
タカにトムがプレゼントをあげる。
Tom will give Taka a present.
-
トムがプレゼントをタカにあげる。
Tom will give Taka a present.
From these three examples, we can see that the order of ‘giver’, ‘recipient’, and ‘object’ does not matter, so long as the correct particles are used.
あげる should not be used when giving something to someone of a higher status, or when lowering oneself to elevate another. This is done through 尊敬語（そんけいご）, or 謙譲語（けんじょうご）, and will use different verbs.
Although
あげる is considered polite speech, it is the most natural way of saying ‘to give’. やる is the ‘casual’ equivalent of
あげる, but due to やる sounding a little bit rough,
あげる became the standard.