They're two slightly different things.

A majority government controls at least half the votes in parliament - so numerically it has a majority over all other parties put together, and as long as its own MPs turn up and don't rebel it can guarantee that votes will go its way. Example: almost all UK governments in recent history, at least immediately following the General Election. By the end of the 1992-1997 Parliament, John Major's Conservatives (who won a narrow majority in 1992) had lost sufficient MPs at byelections that they'd transitioned from being a majority government to a minority.

A minority government, on the other hand, controls less than half of the votes, and so is dependent on at least some opposition MPs either abstaining or supporting them. Example: 2007-2012 SNP administration in Scotland, where they were unable or unwilling to agree terms with any of the other parties to form a coalition, but equally nobody else was prepared to put together a larger coalition either.

In short, a majority government is much more able to push through radical or controversial legislation, while a minority government generally needs to pursue consensus and "not scare the horses".

Coalition is where two (or more) parties agree to band together, normally to achieve a majority.

But, for example, you could've created a Labour/Lib Dem coalition in 2010. That would've had 315 seats, 8 more than the Conservative total of 307. Providing the other parties (DUP's 8, SNP's 6, sundry other minor parties) didn't band together with the Conservatives, they could have taken power as a minority coalition - but would forever need to risk losing legislation unless they could convince other parties to support it, as you (nominally) need 326 MPs for a majority. I say nominally, since Sinn Fein win seats but don't take them, and the Speaker doesn't vote.