If both tire sets are 15 inch, then yes they'll fit.
Where you can start running into problems with factory wheels is if you go to really wide tires.
Most factory wheels are only 7" wide so something like a 12.5 " wide tire can be a tight fit.
Metric tire sizes are like this:
215/70/R17
The 215 is the width, in this case 215mm wide.
The 70 is the sidewall height, expressed as a percentage of the width. So in this case 70% of the 215 width. This is also called the aspect ratio.
The R stands for Radial (the vast majority of tires are now radial construction. There are still some bias ply mudders around)
And the 17 is the rim or wheel diameter, in this case 17 inches.
Usually the size will have either a P (Passenger Car) or LT (Light Truck) designation of the size, like P215/70/17.
LT tires will have a thicker sidewall and ride stiffer than P tires.
HERE is a great tool for comparing tire sizes.