How readily you can change change your date of travel, and how much it will cost you to that, generally depends on the class of ticket that you have bought.
This isn't just a question of "business class or economy class?"; airlines have a host of ticket classes that they sell at different times and to different markets. These classes are usually designated with a letter — C, J, R, D, I are different fare classes for business tickets; Y, H, K, M, L, G, V, S, N, Q, O, E for economy tickets. Sometimes airlines market the different flight classes by quoting different prices labelled "lite", "bargain", "flexible", etc.
The ticket class code isn't highlighted in the booking process because it's technical stuff that the airline and its partners need to know but the passenger mostly doesn't. But, if you want to cancel your flight or change the dates or whatever, it's the ticket class code that determines what options you have. There are no fixed rules or standards about this; each airline sets its own, so one airline might happily reschedule, say, a 'K' ticket on demand for free, while another might not. The only way to find out is to ask your airline what it will cost you to reschedule your ticket. In general, though, the less you paid for your ticket in the first place, the more it will cost you to alter your travel plans later.