Software is anything that is a program. It's what runs the computer (the operating system). It's also your applications (for example, Microsoft Word, Windows Media Player). Hardware is the computer itself, all the physical objects such as the CPU, the RAM, the motherboard, also the keyboard, the monitor. So if someone tells you you need to upgrade your hardware, they may be telling you you need more RAM, a bigger harddrive, a newer graphics card, a better power supply–all these things are hardware. Upgrading your software could be upgrading the operating system (for example, going from Windows XP to Windows Vista) or getting the latest version of an application, or getting the newest drivers for things like your graphics card because you updated your operating system.
Software is not just downloaded programs. Probably the biggest most important piece of software on any system is the Operating System, because the computer is useless without one. (It can be Windows, Mac OS, Linux, Unix, and other more obscure systems you would never use.)