What is the difference between hard and soft Brexit? Is it only economic related affair or does it also pertain to trade and other sectors? Is it only the PM of the UK who decides whether it will be a hard or soft Brexit? Or does the House of Commons and the House of Lords vote on the terms?
A soft Brexit is one with the least change to the current situation. So, it would include becoming a member of the European Economic Area, like Norway. Whilst the UK would no longer be in the EU, there would still be freedom of movement, and compliance to many EU regulations would be required. EEA membership has been ruled out as an option but would represent the softest of 'soft' Brexit.
A hard Brexit is one where the UK disassociates itself completely from all EU institutions. There would be no payment of any sort of 'membership fee', and no agreement where the European Court of Justice is the arbiter.
Those in favour of a soft Brexit are intending to trade by aligning closely with the EU. Those in favour of a hard Brexit point out that the UK now has the power to negotiate its own trade deals, and that it's not in the interest of the EU to erect trade barriers between the EU and the UK (even though they would be a normal consequence of leaving the trade bloc).
The government will negotiate the deal with the European Commission; it's hard to see how it could be otherwise. Parliament has been promised a vote on the resulting agreement but the consequence of it rejecting the deal that has been agreed would be a no-deal Brexit, which would be hard by default.
While the accepted answer is on the face of it correct, there is a slightly different perspective occasionally presented in the media.
For some a hard Brexit is one where the UK is no longer part of the "single market" and "customs union". A soft Brexit is one where the UK belongs to one or other of those two entities (membership of which effectively retains the jurisdiction of the ECJ, freedom of movement of people, and/or the inability to negotiate free trade deals in your own right).
In fact the "single market" is really the EU's internal market, in which goods, services, capital and people can move more or less freely between all countries. (i.e without constraints or tariffs). In practical terms this isn't entirely true but it's close enough.
The "customs union" represents a common customs area in which imported goods from all external countries trading with EU member countries - excepting those with a negotiated free trade agreement - have a tariff or additional charge applied at the point of entry into the country.
While it may seem that tariffs are applied to damage external countries, in fact it is EU consumers who pay in higher priced goods. So food, clothing and footwear all attract tariffs; and imported vehicles attract tariffs of around 11%. These higher prices are paid by consumers to protect internal uncompetitive industries and by extension their employees.