Any 32bit application should run on 64bit too, but if it uses certain library it must also be installed on the new computer. Also you have too make sure the new environment fulfills the requirements of the application, it can be something like OpenGL driver, database engine, the existence of certain file or path, etc.
If the application was built using Lazarus/FPC and you still have the source code, then you can recompile it on the new computer. The first thing you should do is to make sure your Lazarus is working correctly. You can test it by write and compile a simple 'hello world' program.