Example sentences
gāi该shuì睡jué觉le了。
Time for bed.
tā他hái还zài在shuì睡jué觉。
He's still sleeping.
About this word
睡 is "to sleep" and 觉 here means a bout of sleep, so 睡觉 is "to sleep a sleep", the way English says "to dream a dream". Watch the pronunciation: 觉 is jiào in 睡觉 but jué in 觉得 (to feel) and 感觉 (feeling). 睡觉 is a verb-object pair, so other words can slip between the two halves: 睡了一觉 (slept for a while), 睡个好觉 (sleep well). An ordinary verb like 喜欢 cannot be split that way.
The story behind the character
睡
目 (eye) plus 垂 (to droop, also the sound). Eyelids drooping. That's sleep. Xu Shen wrote 睡, 坐寐也: "dozing while sitting."
觉
A shortened 學 on top, 见 (to see) below. From not seeing to seeing. That's waking up, realizing. Xu Shen wrote 觉, 寤也: "to awake."