Can I play old World of Warcraft expansions content on the latest expansions?

Much of the original content (quests, dungeons, raids, etc.) from each of those old expansions is still available and is still playable. Some exceptions exist; for example, the level 60 version of Naxxramas from version 1.11 is no longer accessible within the game, since it was replaced with the Northrend version in Wrath of the Lich King. Many old quests, NPCs, and zones were removed or redesigned completely when Azeroth was reshaped for Cataclysm. Some things have endured unchanged since their implementation, however (such as most of the Outland content) and that content is mostly still perfectly accessible.

However, wide system-level changes have occurred to the game many times over in the intervening time, and those changes apply to all players, regardless of their level or what expansions they own. For example, the talent system has been redesigned several times over the lifespan of the game, and all characters are subject to the latest version of that system; it's impossible to play the game today using the talent system from an earlier period in time. Similarly, more recent feature additions such as pet battles, achievements, and the dungeon finder are accessible to all characters.

As far as "what content can I play with which expansions," you need to own every expansion in order to level up to the maximum level cap. It is not possible to play through all of the previous content all the way up to the current level cap simply by buying Mists of Pandaria and skipping all previous expansions. (Since the base game now automatically includes all expansions up through Mists of Pandaria, though, this point is largely moot.)

Updated for Shadowlands (patch 9.0) If you buy the base game...

• You get the basic races and can unlock the allied races.
• You get all classes, including deathknight, monk and demonhunter.

You get access to all content that is in the game prior to shadowlands to level from 10-50. This includes Burning Crusade (Outlands), Wrath of the Lich King (Northrend), Cataclysm (Eastern Kingdoms and Kalimdor), Mists of Pandaria (Pandaria), Warlords of Draenor (Draenor), Legion (The Broken Isles) and Battle for Azeroth (Kul Tiras and Zandalar).

You get access to a separate installer/server for wow Classic content (Eastern Kingdoms and Kalimdor). To play that content, make a character specifically for that server.

Shadowlands is the current expansion available for purchase. If you buy it then you get level 50-60 Shadowlands content.

You don't "install" expansions; everyone uses the same game client and the same servers regardless of what expansions you have purchased.

Instead, the expansions only unlock the higher level content associated with them.

As per the Blizzard.net shop, if you buy the base game you get all the 1-90 content.

Can be seen here.

Then to get access to the latest Warlords of Draenor, you buy the expansion pack.

Contents can be seen here.

So to answer your question, you only need to buy the base game which gives you all 1-90 content, and WoD which gives you the final 91-100 content. Vanilla, Burning Crusade, Wrath of the Lich King, Cataclysm and Mists of Pandaria are all included in the base game.

Not that a Pandaren Monk is particularly fond of necromancy, but for those people looking at this question at the present time:

All prior expansions to the current expansion of WoW are included in the "base".

For example: at the current moment World of Warcraft: Legion is the current expansion. It requires a prior purchase of World of Warcraft ("base"), which includes all previous expansions (WoD, MoP, Cata, WotLK, TBC, and Vanilla). This trend has continued since at least MoP where all prior expansions to the current one are included in "World of Warcraft" or the "base" game.

You can play all non-current expansions with the WoW purchase. The current expansion requires a separate payment. The nice thing starting with WoD is that Chromie keeps popping up in various places so you can see the same zone at different points in the timeline, meaning that you can see the present day or journey back in the past (at least for zones changed from WoD onward - as of 1/2018 Blasted Lands and Silithus).