Example sentences
tā他chàng唱gē歌hǎo好tīng听。
He sings well.
yì一qǐ起chàng唱gē歌。
Let's sing.
About this word
唱 is "to sing" and 歌 is "song", so 唱歌 is "sing a song". 歌 covers any song and builds 歌手 (singer), 国歌 (national anthem), 歌词 (lyrics). The compound is verb-object, so a song goes inside it: 唱了一首歌, 唱歌唱得很好. When you say you enjoy singing, 唱歌 is the natural choice, since 唱 on its own sounds unfinished without something following it.
The story behind the character
唱
口 (mouth) on the left, with 昌 on the right giving the sound. Xu Shen wrote 唱, 導也: to lead, to start a song and have other voices come in behind you. 昌 also means bright and flourishing, but here it is doing nothing except sounding right. 唱歌 is a mouth plus a sound-alike that happens to mean flourishing.
歌
歌 is 哥 giving the sound, plus 欠, which is a person with an open mouth, breathing out. Xu Shen wrote 歌, 詠也: to chant, to sing the words. 欠 turns up in characters about breath and wanting, like 吹 and 欢. So 歌 is a sound-alike standing next to a mouth letting air out. 唱歌, 歌手, 一首歌.