打 = 扌shǒu (hand radical — the simplified form of 手, hand) + 丁 dīng (a nail; a person's name; a small unit; phonetic). The hand radical anchors 打 in the vast family of hand-action characters: 拿 (to take), 拍 (to clap), 推 (to push), 拉 (to pull), 掉 (to drop), 扔 (to throw). 丁 contributes pronunciation and the image of something pointed — a nail being struck.

The original meaning was narrow: to hit with the hand, to strike. But 打 expanded through spoken Chinese into a generalized light verb — a semantically bleached action word that combines with nouns to form compound expressions. This is the pattern of 打 that makes it one of the most important verbs to learn: not the hitting, but the doing, making, playing, and handling.

A light verb (轻动词 qīng dòngcí) is a verb that contributes little semantic content of its own — it functions as a grammatical carrier that combines with a noun or object to form a complete predicate. English has them too: "make a decision," "take a look," "have a swim." The actual meaning lives in the noun; the verb just activates it.

打 is the most productive light verb in Mandarin — far outstripping its competitors 做 zuò (do/make), 搞 gǎo (do/handle), and 玩 wán (play). The range of 打's combinations is so wide that learners must simply memorize them: 打电话 (make a call), 打针 (give/get an injection), 打包 (pack up / to-go box), 打折 (give a discount), 打招呼 (greet). The underlying logic is often a reaching-and-applying motion — but the extensions have become idiomatic and are not derivable from first principles.