What were the practical effects of the American Constitution of 1788?
There were (and continue to be) multiple practical effects of the writing of the Constitution. The most important of these practical effects was that the Constitution allowed the new country to continue to function as an actual unified country instead of splitting up into a separate little countries.
Before the Constitution was written, the states had tremendous powers under the Articles of Confederation and the central government had very little power. This is not a good recipe for success. If a country has a system like this, it will not remain (de facto, at least) a real country. Instead, it will be a country in name only where the states might as well be independent.
Had it not been for the Constitution, it is likely that the states would have drifted further apart, instead of seeing themselves as part of a greater whole. They had already done this to some extent, doing things like having trade wars and boundary disputes. If this had continued, there would probably never have been the large, powerful, united country that the United States now is.
If by practical, you mean immediately evident to people at the time, I think the most important effect was the ability to govern interstate commerce, which was one of the major reasons the Constitutional Convention was called in the first place. The ability to establish excise taxes was pretty important to people at the time as well, as the Whiskey Rebellion of 1794 made pretty clear. Above all, the Constitution created a central government that was supreme over the states, while leaving room for the proper role of states and federal government to be contested.
The creation of the Constitution had many effects on America. The preceding Articles of the Confederation gave the federal government limited rights. With the creation of the constitution a court system was created to deal with state and citizen issues; and the Federal government was able to create an army. The Constitution created the federal government we have today.