When a person doesn't have Smicha, he can't do anything that requires a Beis Din (a Jewish court).
Therefore, technically we shouldn't be able to
- Force someone to come to court.
- Arrange divorces.
- Arrange conversions.
- Make a new month.
Hillel II (who was an Amora in the middle of the Gemara's time) was mekadesh all Roshei Chodoshim until the coming of Mashiach.
With regards to the other cases which require "real smicha", people with "real smicha" have already appointed us as "emissaries" for certain things. Therefore, we can judge cases that are:
- Involve (monetary) loss.
Cases which are not common (like people punching people) or don't involve monetary loss (like fines) are impossible to judge nowadays. [Even though this is true technically, a court has the right to go beyond the law if they must. Since there are no truly ordained Rabbis nowadays, they would judge all types damages and thefts).
The Aruch Hashulchan says that conversions and divorces are considered like something which "causes a loss" as people would be prevented from coming under the wings of the Shchina, and women would be stuck (respectively).