What is international border and how it is different from local borders?

In international border is a border between two independent countries.

Local borders are the boundaries between sub-divisions of a country, for example between federal states, provinces, counties, municipalities.

At an international border the laws of one country end, and those of another begin. The authority of one government ends, and that of another starts.

You need the permission of both governments to cross the border. There will probably be an identity check, for which you usually need a passport. Any goods you are carrying may be inspected and you may be charged customs duties on them, or they may simply be refused entry. If you are travelling in your own vehicle, then that vehicle has to be legal in the country you entering, and your driving licence has also to be valid in that country.