Malware is an abbreviated term for "malicious software".  It refers to any software that is designed to specifically disrupt or damage a system.  Malware includes such things as worms, Trojan horse, and viruses.

Trojan Horses

A Trojan horse is a program that appears to be useful or at the very least harmless.  However, it has been designed to contain hidden code to exploit or damage a computer system. A Trojan horse neither replicates nor copies itself, but performs some illicit activity when it is run. It stays in the computer doing its damage or allows somebody from a remote site to take control of the computer.

There are other terms that are associated with Trojan Horses: Remote Access Trojans, and Rootkits.  Remote Access Trojans are commonly referred as backdoors.  They allow someone else to control your computer from a remote location.  A rootkit is a collection of software programs that once installed allow someone to gain unauthorized remote access to your computer

Worm

A self-contained program (or set of programs) that is able to spread copies of itself to other computer systems.  A worm can consume network or local system resources.  They can cause a denial of service attack.  A worm may also deliver other malware such as keyloggers in addition to spreading itself.

Virus

A virus is a program code that can cause damage to hardware, software or data. Virus code is usually buried within the code of another program (file, document or boot sector of a disk) and once executed it will attempt to replicate itself by infecting other hosts across the network.  Some times viruses are used to deliver other types of malware such as a Trojan horse.

How Malware spreads

There are many different methods that malware will attempt to use to infect computer systems:

Remote vulnerability exploit:     This occurs when a piece of malware attempts to gain access to a computer system by exploiting vulnerability in a service or an application.  This is seen in worms.

Email:    Email is an easy way for attackers to transport Malware to hundreds of thousands of people. It is very easy to trick users into opening e-mails or opening e-mail attachments using social engineering techniques.

Network Shares:   Network files systems with poorly implemented security produces an environment where malware can spread to a large number of computer.

Removable media:   This means of transport started with floppy disk and has spread to such devices as Universal Serial Bus(USB) and Firewire devices.

Network scanning   : Malware writer use this mechanism to scan networks for vulnerable computers that are susceptible to remote exploits. 

Peer-to-peer (P2P) networks  - Users install a client component that makes their file system accessible to other computers across the network.