At the minute if you travel between ROI and NI the only clue you're in a different country is the speed limit signs change (kph in ROI, mph in NI).
This is because there is a fully open border for people and goods.
Post Brexit that won't necessarily be the case. People probably won't be a problem because of the Common Travel Area between ROI and the UK (among others). This isn't actually legally confirmed, but it's the case in all likelihood.
However goods are more complicated. If the UK were to stay in the European single market and the customs union, things could continue as they are. This was the soft Brexit you may remember hearing about.
However the UK have decided not to do that. That means at some point there will be different regulations (for example for agriculture) on different sides of the border once the UK decides to deviate from the EU.
The future relationship hasn't actually been decided yet, that's what the two year transition period is for. The backstop originally stated that if the future relationship would necessitate a border, or if no agreement was reached, NI would effectively remain in the single market and customs union (think it's legally a bit more complicated than that, but that's the gist) allowing the border to remain open. This would effectively put a border for goods in the Irish sea, between the island of Ireland and GB.
The UK didn't like the idea of a border between NI and GB (namely, the DUP didn't like it) and so the EU made a huge concession and expanded the backstop to include the whole UK.
If the UK crashes out without a deal, the backstop won't apply and the likelihood is a return to a hard border at some point in the future - maybe not day 1, but it's naive to think it won't happen.
Putting up the border is impractical. There's a couple of hundred crossings, including through people's houses, businesses etc. People work on one side and live on the other, it'd be like putting an international border and customs post between you and your local Tesco. When this was the case before, the British government actually blew up some roads so they couldn't be used as crossings.
And then you have the whole emotional problem of the border, which would take another post just as long and my train is pulling into the station... Suffice to say, a border would mean Bad Things for the peace process.