When you suspire, you breathe deeply or sigh. You might suspire melodramatically as you listen to your best friend complain about her allowance for the thousandth time.
The verb suspire is considered obsolete today—you'll probably only encounter it when you're reading poetry. In Robert Frost's poem "Sitting by a Bush in Broad Sunlight," he wrote: "And from that one intake of fire / All creatures still warmly suspire." Not only is it a literary way to say "breathe," but it also rhymes nicely with "fire." The Latin root is spirare, "to breathe."