Several factors come into play (this is just a list of what I know, a lot of things I dont know )
The big things are:
1. A certification that the plant is not a wild collected endangered species, otherwise it needs a proper CITES permit. Commercially propagated plants are CITES exempt I think, or the CITES permit just states that the plant is commercially propagated.
2. Phytosanitary clearance:
This is a certificate that declares that the plant does not harbor any pests and diseases that could be a threat to the receiving country/state. In the USA, different states have different phytosanitary clearance requirements since each state has its own list of "commercially significant pests & diseases". For example, one pest may be a serious threat in Hawaii, but is not an issue in Minnesota because the pest cannot survive below 40F.
This is one of the reasons why some online orchid vendors do not ship plants to certain states in the US because the process of acquiring the phyto permit is sometimes too tedious and is cost prohibitive.
3. Invasive species clearance:
Some countries/states also have a list of foreign plants that are prohibited because it competes with native wildlife if they escape ("Houseplants gone wild" as they say in Hawaii).
I am not sure of the actual process for this one, probably need to check with the local agriculture department.
Again, this is one of the reasons why certain vendors cannot ship specific plants to certain states in the US.
4. Embargo conditions (this is probably very rare).
Some countries have imposed embargos on other countries that prohibit certain levels of trade between the countries. Again, this is rare but is still worth checking.