You will likely find this old response of great use.

To give a TL;DR summary of what I wrote there though.

The Mujaheddin refers to the Afghan fighters who fought against the Soviets during the '80s.

Al Qaeda is an organization founded by Osama bin Laden in the late '80s. Its genesis is with the foreign fighters who traveled to Afghanistan to help the Mujaheddin.

The Taliban was founded in the early '90s, when the Mujaheddin, having kicked out the Soviets and taken nominal control of the country, was now fighting itself.

While former Mujaheddin fighters certainly ended up joining the Taliban and Al Qaeda, neither is a direct successor organization to the Muj from the '80s.

That old response linked by /u/Georgy_K_Zhukov is awesome. I just wanted to add a little bit about the names of these orgs because it gives further insight:

• Mujahideen is a plural of mujahid which refers to a practitioner of jihad, which is Arabic for "struggle" or, as is commonly understood by Muslims, resistance against oppression, whether that oppression be that of the self upon the soul or an external political or armed entity. The latter meaning is commonly attributed to in the West and, as an example, the Mujahideen of Afghanistan believed that they were committing jihad against the non-Islamic oppressive forces arrayed against them, both Soviet and Afghan.

• Taliban is a plural of talib which is Arabic/Pashto/Urdu/Punjabi for "student". The name is derived from their story of outraged Afghan refugee students from the madrassas of Pakistan committing jihad against sexually deviant warlords. These madrassas were funded by wealthy Arab, esp. Saudi, donors, and followed Wahabbi/Salafi doctrines. Just as an intellectual exercise, and hopefully, not violating the 20-year rule, draw the parallel to the current situation in Syria.

• Al-Qaeda means "the foundation" though, more colloquially in that region, the term is also used to refer to the textbook used in class which contains all the lessons for the year.

The names reflect objectives and tactics.

• Mujahideen were loosely-aligned bands of Afghan guerrillas that received sophisticated weaponry from client states and morphed into warlordism. These were legitimized as freedom fighters by President Reagan and thus, earn the capital "M".
• Taliban were young, brain-washed students recruited from schools in the refugee camps of Pakistan and formed a cohesive army, not unlike the current form of ISIS, led by a mythical lord who evoked the legends of the end of times. His election by acclamation as amir-ul-momineen i.e. "leader of the believers" is a title that was last held by the last of the Righteous Caliphs, Ali, so it was quite a scandal when Mullah Omar declared it for himself, despite his use of relics of Prophet Muhammad to present his case for the title.
• Al-Qaeda is less of an army and more of a political movement. Its choice of terrorism is a political tool to invite disproportionate response and turn the otherwise unaligned populace against what it believes to be oppressors. It funds ideologically-aligned but loosely-coupled jihadi/mujahideen groups (in the West, due to President Reagan's blessings, "Mujahideen" has acquired a positive connotation while the term "jihadi" refers to terrorists and their funders).

Hope this helps.