What is the difference between the terms "mass murder", "massacre", and "genocide"?

The difference between manslaughter and murder is usually a legal one, and depends on which jurisdiction you are in.

In everyday lingo, manslaughter is usually seen as the involuntary or unintended ending of a life, whereas murder implies intent: a murderer meant to end a person's life. Also, the word implies that the killing is illegal if we call it murder.

Killing is a neutral term, which does not imply that the acting party is necessarily at fault. Context often gives more detail.

So, a mass killing is the active, but not necessarily malicious, ending of many human lives. This could be used, for instance, in the case of the Spanish Flu, which possibly killed more people in the years after WWI than the Great War itself had claimed.

Mass murder is the intentional killing of many humans, either by one person, a group of people or an organisation. It does usually refer to a single event, so a regime routinely killing citizens would not be accused of mass murder. The same regime killing a lot of people at a single event would be.

A massacre is similar to mass murder, but can be applied in cases where murder is not deemed applicable, such as in a war situation. In war, killing is normally not seen as murder (murder being intrinsically negative and/or illegal). A massacre in war is also not always morally wrong, although it will have negative connotations in cases involving non-combatant victims.

Genocide is the killing of people based on race or ethnic background. All the other words have no such implication whatsoever, so that sets genocide clearly apart from all the rest. Also, genocide and other -cides can take place over a long period of time, involving many individual killings (whereas massacres usually refer to a single event where many people die - a genocide can include one or more massacres.)

The -cide suffix is used productively to describe the (systematic) killing of individuals based on other criteria as well. For example, the preference for male children in some parts of the world has led to what some people refer to as gendercide, the killing of people (mostly children or babies) based on sex.

Genocide carries a distinct connotation of being directed towards a particular group of people (or peoples):

The systematic and widespread extermination or attempted extermination of a national, racial, religious, or ethnic group. - TFD

It comes from the Greek: géno(s), "race" + -cide: "kill"

The Holocaust targeted Jews; also persecuted were Romanis, Christian Poles, communists, homosexuals, and others. The name comes from the Greek: hólos, "whole" and kaustós, "burnt".

In Cambodia, the mass murder committed by the Khmer Rouge is also considered a genocide, in this case a class of people were targeted: the educated.

All genocides are mass murders, but not all mass murders are genocides.

the savage and excessive killing of many people; carnage, massacre, slaughter, butchery - TFD

Mass murder is often used to denote smaller numbers of killings in a short amount of time, for example, the killing of over 70 people in Nuevo León, Mexico on 25 June, 2010, or the people killed at a theater in Aurora, Colorado.

All are murders. Murder is defined by the law (and by moral law.)

The killing of another person without justification or excuse, especially the crime of killing a person with malice aforethought or with recklessness manifesting extreme indifference to the value of human life.

Homicide is the killing of any human by another:

The killing of one person by another, regardless of intention or legality.

It is derived from the Latin homicīdium: homō, man, and -cīda, kill.

Manslaughter is legally-defined killing of a particular sort. A dictionary will help you with the particulars.

The killing of a person without malice aforethought but with either the intention to commit an unlawful act that leads to an unintended death, or with an otherwise murderous intent that is extenuated by some partial defense, such as acting under the influence of an extreme emotional disturbance occasioned by a substantial provocation on the part of the victim. - TFD

The international legal definition of the crime of genocide is found in Articles II and III of the 1948 Convention on the Prevention and Punishment of Genocide. Refer to this article.

Legally, Genocide is any of the following acts committed with intent to destroy, in whole or in part, a national, ethnical, racial or religious group, as such:

(a) Killing members of the group;

(b) Causing serious bodily or mental harm to members of the group;

(c) Deliberately inflicting on the group conditions of life calculated to bring about its physical destruction in whole or in part;

(d) Imposing measures intended to prevent births within the group;

(e) Forcibly transferring children of the group to another group.

The Genocide Convention does more than define the crime, it holds states responsible for preventing and punishing genocide. This obligation is stated in Article 1 of the convention under which states must prevent or punish genocide, individually or through the United Nations, and the International Criminal Court can be involved.