A sickbed is the bed an ill person is lying in — but the word carries more than "a bed". It is used when the illness is serious and long, and when other people are gathered around: you rise from your sickbed, you are called to someone's sickbed, a family keeps watch at a sickbed. It is a slightly old-fashioned and rather literary word, common in novels, obituaries and news writing, and rare in ordinary speech. If you have a cold, you say "I'm in bed", not "I'm on my sickbed" — unless you are joking about how sorry for yourself you feel, which people do quite often. There is also a figurative use in politics and business writing: a company or an economy can be described as being on its sickbed, meaning weak and in trouble. Note the prepositions: usually on his sickbed or at her sickbed (standing beside it), and from his sickbed for someone who gets up despite being ill.