Short answer
American Conservatism began as an ideological reaction to The New Deal, which, while politically successful for Democrats, was an economic and social disaster. It prolonged the Great Depression until WWII put the country back to work, whereas the major effects of the crash were dissipating in Europe halfway through the decade; and it began the myth that Democrats were the party that cared about the black population. It also sounded the alarm, for those listening, about the folly of expanded government as a social engineering project.

Neo-Conservatism was born on the left. It may sound like an intentional case of cognitive dissonance but there really was a right wing of the Socialist Party. It was anti-Stalinist, and pro-American, up to a point, but it identified itself more in terms of what it was not—radical socialists and Communists. The New Left of the 60’s likewise distinguished itself from the old 30’s style socialists, but in a way that sharpened the divide for such intellectuals (New York City Jewish Intellectuals, actually, a real thing, with a Wikipedia page and everything) as Irving Kristol, and Norman Podheretz, who found themselves more in alignment with conservative Democrats (more cognitive dissonance) like Henry Jackson. They made the trek to the conservative camp in the 60’s and 70’s, where they still thrive through such spawn as Bill Kristol and John Podheretz, both confirmed never Trumpers

Neo-cons have never been fully welcomed by traditional Conservatives, who trace their current origins to Russell Kirk’s “The Conervative Mind,” 1953, and William F. Buckley’s National Review, which began publication in 1955. Nine years later they wrested control of the Republican Party from the Rockefrller moderates and went down in flames with Barry Goldwater, and fifteen years after that rode into town with Reagan.

Alt-Right is bullshit. It’s a label that has been co-opted by the media/left to slime anyone to the right of Elizabeth Warren. For a time, Milo used it to brand a style of conservatism characterized by his intentional buffoonery and theatrics as a means of ridiculing all things PC and left, blended with the media activism of Andrew Breitbart. It would probably have endured more if he’d just used “New Right.” But Breitbart is dead (of a heart attack on a sunny day, shortly after he accused John Podesta of pedophilia in a public forum), and Milo’s been cancelled and we’ve once again let the left define the terms of the debate. We really need to stop doing that.

A conservative is a centrist American who supports the U.S. Constitution and celebrates America's founding principles of Life, Liberty and the Pursuit of Happiness.

A Neo-conservative is a globalist who supports foreign intervention and nation building. These are typically represented by Republican politicians who defer domestic political power to their Democrat Party colleagues.

The Alt-right are leftists who perpetrate despicable acts to smear conservatives. This is a political class wholly disassociated with the conservative movement and the Republican Party.

You don't. They are all left wing labels to try to divide the people. They love to label everyone into seperate groups in order to divide and conquor. Don't fall for their propaganda.

Conservative means what it means.
Neo-conservative sometimes referred to as neo-cons, neo-conservative
relating to or denoting a return to a modified form of a traditional viewpoint, in particular a political ideology characterized by an emphasis on free-market capitalism and an interventionist foreign policy.
Alt right- an ideological grouping associated with extreme conservative or reactionary viewpoints, characterized by a rejection of mainstream politics and by the use of online media to disseminate deliberately controversial content. Usually conspiracy theorists are alt right, so are the memers groups.
Summed up- there doesn't appear to be much difference with regular conservatives and neo-cons, while the alt right could be considered a lunatic fringe (one of the two known political extremes). I mostly identify as a conservative-leaning libertarian, meaning most of my political views lean to the right while I do agree with a few liberal views. Difference between liberal and progressive is a whole other topic perhaps for another post another time. Hope this was helpful.