Oblivion:Disposition

Disposition is an attribute that describes how much an NPC or Creature likes your character, or any other NPC or creature.  In-game displayed value ranges from 0 to 100, while the actual value can be below 0 and above 100.  For friendly NPCs, the value of their disposition towards you can be seen from within the Speechcraft or Bartering dialogues; for other NPCs and creatures, it can only be seen in-game using the Console (for example, via the SetDebugText options).  Many factors will affect an NPC's disposition during the game.  NPCs remember their disposition towards you from one conversation to the next (although it may change in the interim, for example, if your Fame increases).

Effects of Disposition
Disposition affects many actions that NPCs take:
Disposition improves the prices you get from merchants by haggling.
Often NPCs will not tell you certain information until their disposition is greater than a certain value (the exact value depends upon the character and the quest).
If a guard has a disposition of at least 91 towards you, he will not arrest you, but instead will pay your bounty himself (as long as your bounty is less than 1000 gold).  
Disposition alone does not determine whether an NPC or creature will attack you. If their disposition towards you is higher than their Aggression score, then they will not attack. Enemies are programmed to have negative dispositions towards you.  However, if you charm them enough (typically by 90 or more points) they will no longer attack.  If charming an intelligent creature, you can talk to them, ask about rumors, and even use bribes and the speechcraft mini-game to permanently increase their disposition towards you.  With enough work, you can turn them into your friend.  This even works with Dremora (although their conversations are fairly limited).
NPCs with high disposition towards you will come to your aid in battles, so long as they physically see you fighting. For example, an NPC who likes you will fight the guards when you resist arrest. 
If a beggar's disposition towards you is at least 70, they will tell you information for Thieves Guild quests without you having to make a donation.
Most NPCs will refuse to talk to a full (i.e.100%) vampire unless their disposition is 100.
NPC greetings and many other dialogues will depend upon the NPC's disposition, with friendly greetings at high disposition and impatient or even insulting greetings at low disposition. Example: if a guard's disposition to you is 50 or higher, when you are sent to jail he will say: "Serve your time peaceably, and pay your debt to society."  Whereas if it is less than 50, he will say: "Hope you rot, criminal scum."

Starting Disposition
An NPC's starting disposition toward the player consists of a number of factors:
Base disposition is determined by both the NPC's and the player's Personality score. If both the player and the NPC have the same Personality, then the base disposition is equal to that score, i.e., if both the player and the NPC have a Personality of 40 (and no other modifiers apply) the starting disposition will be 40. For every 4 points of Personality that the player has above the NPC's Personality, the starting disposition increases by 1 point. If the player's Personality is lower than the NPC's, the disposition is decreased by 1 point plus 1 additional point for every 4 additional points of Personality below the NPC's Personality, i.e., if the player has a Personality of 39 and the NPC a Personality of 40, a -1 disposition modifier is applied. If the player's Personality is 35, a -2 modifier is applied, etc.
The base disposition is modified by Race reactions.
It is also modified by faction reactions. In general, joining a faction will raise the disposition of other guild members by an amount determined by the faction. These disposition modifiers accumulate with each rank you attain in a guild. (See specific factions for details.) NPCs may also be members of 'invisible' factions which do not form an official guild but which represent fans or enemies of the player. These factions have a set disposition modifier toward the player (or Player Faction).
Fame and Infamy also affect disposition. Although these calculations are more complex (see the appropriate pages for details), essentially, most NPC's dispositions will increase at a rate of 3 points for every 10 points of Fame, and decrease at a rate of 3 points for every 10 points of Infamy. (See the note below under Increasing Disposition.)
Circumstantial factors may also apply, such as speaking to an NPC with a weapon drawn, or committing a crime that the NPC is aware of. (See Decreasing Disposition below.)
Finally, disposition may be affected by any Charm effects active on the NPC.
The game uses the same formula to determine inter-NPC dispositions. (Though, NPCs do not have fame or infamy.)

Increasing Disposition
There are many ways to increase disposition:
Disposition is increased by Personality: every 4 point boost in Personality increases NPC dispositions by 1 point.  This benefit applies to Personality increases past 100.
Joining guilds or other factions increases the disposition of members of that guild/faction.  Each promotion within the guild will further increase the members' disposition.  The magnitude of the disposition increase depends upon the faction (+3 per rank for Thieves Guild, +10 per rank for Fighters Guild, +20 per rank for Mages Guild, +31 per rank for Dark Brotherhood).
The Speechcraft minigame can be used to increase Disposition, but only up to a predetermined maximum value depending on your Speechcraft skill (e.g. 79 at skill level 50).
  can be used, but they become more expensive and less efficient as the disposition goes up.
The equations that govern bribery are detailed at  
Buying or selling an item, or a pack of items (e.g. 50 arrows for 2 gold each), worth 100 or more gold will increase a merchant's disposition by 1 point each time.
NPC dispositions will go up as you become more famous.  Dispositions will increase 3 points for every 10 Fame points you acquire, up to a maximum bonus of +20 disposition (at 67 Fame).
The dispositions of a few NPCs will also go up as you become more infamous.  The effect of infamy on NPCs depends upon the NPC's Responsibility.  Although for most NPCs, infamy decreases disposition, for NPCs with particularly low responsibility (e.g., 10), infamy will instead increase disposition, up to a maximum bonus +20 disposition.
''Charm'' is very effective, and typically only needs to be a very brief effect (just long enough for you to start talking to the NPC).
In addition to the Charm spells that can be purchased, it is possible to acquire a free (but magicka-expensive) Captivate spell by completing the Bravil Recommendation quest.

If you want to permanently raise an NPC's disposition to above 100 without bribing them you need to:
# Create a custom Instant Speechcraft spell: Fortify Speechcraft 100 points for 2 seconds on self.
# Temporarily lower the NPC's disposition towards you by 10 points by drawing your weapon or fists.
# Cast the Instant Speechcraft spell and immediately talk to the NPC (before the spell expires).
# Play the disposition minigame until his disposition is maxed out (usually above 90 points).
# End the conversation and sheathe your weapon.
# Talk again to the NPC, and his disposition will now permanently be at (or above) 100.

Decreasing Disposition
You may also find that you want to decrease someone's disposition.  This is particularly true late in the game if you want to increase your Speechcraft skill: if you are famous enough, everyone's disposition will already be maxed and you will have a hard time starting the Speechcraft minigame.
Talking to someone with a weapon in your hand (or even just with your fists) will temporarily lower their disposition by 10 points. If you sheathe your weapon and talk to them again, the NPC's disposition towards you will be back to its normal value.
Any method of temporarily decreasing your Personality attribute (e.g. drinking Cheap Wine or using a custom Drain Attribute spell) will also decrease your disposition score with all NPCs.
Any merchant's disposition can be lowered by pushing the Mercantile haggle slider to the far right and repeatedly trying to buy or sell an item.  Each time they refuse your offer, their disposition will go down by 1 point.
Commit a crime in sight of the NPC.  The magnitude of the effect depends upon the nature of the crime (assault: -10, trespass: -20, pickpocket: -25, murder: -50, stealing: -0.5*value in gold of the stolen item(s)). A crime committed against the witnessing NPC will have twice the effect of a crime committed against someone else (although the target of the crime must be someone liked by the witnessing NPC to have any effect).
Joining guilds and being promoted within those guilds will decrease the disposition of certain NPCs.  In particular, guards have a -3/rank disposition modifier towards Thieves Guild members (but there is no such penalty for Dark Brotherhood membership).  There are also a few specific NPCs who dislike certain guilds.  One example is Ontus Vanin, who has a -5/rank disposition modifier towards Mages Guild members.
Most NPC dispositions will decrease as you become more infamous.  The disposition penalty depends upon the NPC responsibility.  For NPCs with high responsibility (i.e., 100), you may lose 1 disposition point for every 2 points of infamy.  For NPCs with low responsibility (i.e., 25), you may only lose 1 disposition point for every 16 points of infamy.  At even lower responsibility (i.e, 10), you will instead gain disposition from infamy.  In all cases, the maximum disposition decrease is -20. 
Casting a Command Humanoid spell on an NPC will cause the NPC's disposition to decrease when the spell wears off.
Be cautious when decreasing the disposition of NPCs.  If decreased too low (e.g., to 20 or lower), even normally friendly NPCs may attack you, depending upon their Aggression score.