Malware is a malicious software made to collect sensitive data, access secured systems or disrupt operations. It is a collective term for spyware, adware, worms, virus, trojan horses and any intrusive, hostile or nuisance programs. This malicious program would normally be disguised as or comes along with legit software. It is not uncommon for safe programs to be bundled with malware so that a user will unknowingly install it into a system.
The basic characteristics of a malware is that it must be hidden and run without being seen or deleted. Any program with malicious code that can trick users into installing and running it is considered a malware.
Trojan horses are commonly known in their disguise as something beneficial or normal so users unwittingly install them. They are commonly bundled with legit software that can be downloaded online and users install them not knowing that a harmful program is being installed at the same time.
Classified as a malware, virus is a software that will replicate and spread among all other executable files in a system. It should not be confused with the worm because the latter is capable of automatically transferring itself across the network in order to infect other systems.
Rootkits usually act through altering the registry of the operating system itself in order to stay hidden and keep the other malware processes from being displayed in the process list. They also secure malware files by locking them (so it won't be deleted) or foiling attempts to kill the malicious processes by replicating them quickly.
Backdoors work through bypassing the usual authentication system and gain remote access to it while keeping under the radar. And once a system is penetrated, other backdoors could be installed for easier access.