Kill, slay, murder, assassinate, dispatch, execute are comparable when meaning to deprive of life or to put to death.
Kill is so general that it merely states the fact and does not, except in special phrases (as "Thou shalt not kill "), suggest human agency or the means of death or the conditions attending the putting to death. Also, the object of the action may be not only a person or other living thing but also an inanimate or immaterial thing with qualities suggestive of life.
Slay implies killing by force or in wantonness; it is rare in spoken English, but it often occurs in written English where it may convey a dramatic quality whether in poetic or elevated writing or in journalese. In its extended uses slay usually suggests wanton or deliberate destruction or annihilation.
Murder definitely implies a motive and, often, premeditation and imputes to the act a criminal character; it is the exact word to use in reference to one person killing another either in passion or in cold blood. It is sometimes used in place of kill as more expressive or in place of slay as more brutally direct and condemnatory, both in literal and extended use.
Assassinate implies murder especially of a person in governmental or political power by stealth or treachery and often by an agent or hireling of an opposition. It usually suggests an attempt to get rid of a person who is believed to be an obstacle to the safety of a tyrant, the welfare of a people, the liberty of a nation, or the success of a design.
Dispatch also suggests an attempt to get rid of a person by killing him, but it is far more colorless than assassinate. Because it nearly always implies taking direct means of killing (as by shooting or stabbing) and so sometimes connotes expedition or speed in killing or in ending suffering, it is applicable, as most of the other terms are not, to killing (as of a sick or injured animal) for humane reasons. Often dispatch is merely a euphemism for another of the terms of this group when quick killing or a sudden end is implied.
Execute is the term for putting to death one who has been condemned to such a fate by a legal or military process, or sometimes by summary action of a group.