Accident: An unfortunate incident that happens unexpectedly and unintentionally, typically resulting in damage or injury. (oxforddictionaries.com)
Accident came into English in the 14th Century, and originally only meant an incident, something that happened by chance, not necessarily something that was a mistake or negative. It entered English via Old French accident which is derived from the Latin accidentem, which is the accusative form of accidens. That word meant an occurrence or misfortune, and is the noun form of the verb accidere meaning “to happen”.
Accidere is taken from the verb cado meaning “I fall” and the prefix ad- meaning towards. In Latin accidere can be a euphemism for “to die”.
Cado carries on the meaning of death into English in some forms. The English cadaver derived from it. Cadaver is found as early the 3rd century in Tertullian, an early Christian writer from Carthage who wrote in Latin. Cadaver has a folk etymology, supposedly from latin “caro data vermibus” but is almost certainly false.
Another word derived from the Latin cado is decay. Found as early as the 15th Century and taken from Old North French decair meaning to fall away or decay, related to Modern French déchoir, meaning to wane or to strip. The Latin prefix de- means away from or down, so decidere means to fall down, or to die.
A similar word to accident, incident too is derived from cadere, using the Latin prefix in. Found as early as the 15th century, making it a later word than accident, and similarly from the Old French incident, from Latin incidentem, accusative form of the present participle of incidere, from in and cado. Incidere means to fall in or to find the way or fall upon.
A less obvious related word to cado is occasion. First found in the 14th Century, but the sense is quite similar, meaning an occurrence or something happening. Similarly via Old French ochaison meaning a cause, reason, pretext or opportunity, and that from the Latin occasionem, the accusative form of occasio, itself from occidere, meaning to fall down, from ob- and of course cado. Ob is a prefix meaning across or against, but also down or toward. Naturally occido is also a euphemism for “I die”.
Sources: oxforddictionaries.com, http://www.etymonline.com, wiktionary.org, Oxford dictionary of Word Origins Second Edition.