虾 is the general word. Specific types: 大虾 (prawn), 虾米 (small dried shrimp), 龙虾 (lobster, literally dragon shrimp), 虾仁 (peeled shrimp).
register
There is an old reading há, found only in the variant spelling 虾蟆 (a toad), which is usually written 蛤蟆 today. For the shrimp sense the character is always read xiā.
Insect and small-creature radical on the left, historically used for any small invertebrate, including aquatic ones. Anchors 虾 in the small-creature family alongside 蝴 (butterfly), 蛇 (snake), 蟹 (crab), 蜗 (snail). Tells the reader at a glance: this is a small, segmented animal.
Right side 下 supplies the sound, xià to xiā with only a tone shift, an exact match. The traditional 蝦 used 叚 as the phonetic; the simplified form swapped in 下 for brevity. The down meaning is inert here, a pure phonetic borrow chosen for its compact shape.