When you provide a path, it can be absolute/rooted, or relative. If you provide a relative path, it will be resolved by taking the working directory of the running process.
Example:
string text = File.ReadAllText("Some\\Path.txt"); // relative path
The above code has the same effect as the following:
string text = File.ReadAllText(
Path.Combine(Environment.CurrentDirectory, "Some\\Path.txt"));
If you have files that are always going to be in the same location relative to your application, just include a relative path to them, and they should resolve correctly on different computers.