Difficult Terrain is defined in the Combat section as -

Difficult Terrain

Difficult terrain, such as heavy undergrowth, broken ground, or steep stairs, hampers movement.

So, I would say an Acrobatics check would enable you to move at full speed through SOME Difficult Terrain, but not all. On Broken Ground, or some sort of uneven shifting Difficult Terrain, yes, Acrobatics would help. But through Undergrowth or on Steep Stairs? I would say it would not help.

Alternatively, you could use an Acrobatics check to just jump over some Difficult Terrain, allowing you to potentially move at full speed.

The DM will probably be the ultimate authority about

Using the Acrobatics Skill to Avoid Attacks of Opportunity

The rules are contradictory. In one part of the description of the skill Acrobatics, an Acrobatics skill check is made to move through a threatened square (i.e. roll once per creature per square), while elsewhere in the skill description an Acrobatics skill check is made to move through a threatened area (i.e. roll once per creature per threatened area). Luckily, the FAQ clarifies this to mean threatened area, otherwise that could be a lot of die rolls.

However, complicating matters is that the Acrobatics skill check to move through an enemy's space seems to be intended as a separate Acrobatics skill check, but the FAQ believes that moving through an enemy's space can only occur simultaneously with provoking an attack of opportunity for normal movement, ignoring the possibility of enemies with reach.

The examples below, then, are predicated on the following:

• That, as per the FAQ, an Acrobatics skill check is made to move through a creature's threatened area not through each square the creature threatens.
• That, as per the Acrobatics skill and the Tumbling section of Moving Through a Square, a separate Acrobatics skill check is made to move through a creature's space.

The examples below assume Medium-size creatures, none threatening squares beyond those adjacent to their spaces.

Example 1: By One Foe

A wants to reach point o, but B doesn't want A to reach point o.

Not Using the Skill Acrobatics

Fig. 1                         Fig. 2
--A--  A takes a move action.  -----  A enters B's threatened area.
-----                          --A--  This doesn't provoke an attack
-B---                          -B---  of opportunity.
-----                          -----
--o--                          --o--

Fig. 3                         Fig. 4
-----  A moves within B's      -----  A continues his move action. His
-----  threatened area, pro-   -----  movement has already provoked
-BA--  voking an attack of     -B---  an attack of opportunity from B
-----  opportunity.            --A--  so the movement doesn't pro-
--o--                          --o--  voke again from B.

Fig. 5
-----  A exits B's threat-
-----  ened area as part of
-B---  A's move action, still
-----  not provoking again
--A--  for this movement.

Using the Skill Acrobatics
To avoid the attack of opportunity for leaving a threatened square during Figure 3 A makes an Acrobatics skill check to Move Through a Threatened Area.

• Failure means A provokes an attack of opportunity from B. A continues taking his move action, provoking no further attacks of opportunity for normal movement from B.
• Success means A doesn't provoke an attack of opportunity from B but A's normal movement is halved while A continues his move action in B's threatened area.

Example 2: Through One Foe

A wants to reach point o, but B doesn't want A to reach point o.

Fig. 1                         Fig. 2
--A--  A takes a move action.  -----  A enters B's threatened area. This
-----                          --A--  doesn't provoke attacks of
--B--                          --B--  opportunity.
-----                          -----
--o--                          --o--

Fig. 3                         Fig. 4
-----  A must succeed on an    -----  If the Acrobatics check to Move Through
-----  Acrobatics skill check  -----  an Enemy's Space is successful A con-
--A--  to move through B's     --B--  tinues his move action. His movement
-----  space. Afterward A      --A--  has already provoked an attack of op-
--o--  provokes an attack      --o--  portunity from B so this movement 
       of opportunity for             doesn't provoke again from B.
       moving within B's
       threatened area.

Fig. 5
-----  A exits B's threat-
-----  ened area as part of
--B--  A's move action, still
-----  not provoking an attack
--A--  of opportunity again
       for this movement.

Using the Skill Acrobatics
During Figure 3 A must make an Acrobatics skill check to Move Through an Enemy's Space.

• Failure means A doesn't move into B's space, the move action ends, and A provokes an attack of opportunity from B. A continues his turn with the battlefield looking like it did in Figure 2.

• Success means A continues his movement through B's space. To avoid the attack of opportunity for leaving a threatened square during Figure 3 A makes an Acrobatics skill check to Move Through a Threatened Area. B is not an additional opponent, therefore the Acrobatics skill check DC remains unchanged even if it was previously successful versus B (see Table: Acrobatics DC's to Move Through Threatened Areas).

  • Failure means A provokes an attack of opportunity from B. A continues taking his move action, provoking no further attacks of opportunity for normal movement from B.
  • Success means A doesn't provoke an attack of opportunity from B but A's normal movement is halved while A continues his move action in B's threatened area. However, A's movement while in B's space probably isn't reduced because most creatures don't threaten their own space (i.e. from the section Threatened Area under Attacks of Opportunity: "You threaten all squares into which you can make a melee attack,... that means everything in all squares adjacent to your space.")

Note: Most foes will only be able to make one attack of opportunity each round, but even if the foe's incapable of making the attack of opportunity the foe's space still prevents the creature's movement unless the foe's helpless, the creature is 3 or more Size categories smaller than his foe, or the creature's Size is Fine, Diminutive, or Tiny.

Example 3: By Two Foes

A wants to reach point o, but B and C don't want A to reach point o.

Not Using the Skill Acrobatics

Fig. 1                         Fig. 2
--A--  A takes a move action.  -----  A enters B's and C's threatened areas.
-----                          --A--  This doesn't provoke attacks of
-B-C-                          -B-C-  opportunity.
-----                          -----
--o--                          --o--

Fig. 3                         Fig. 4
-----  A moves within B's and  -----  A continues his move action. His
-----  C's threatened areas,   -----  movement has already provoked
-BAC-  provoking an attack of  -B-C-  an attack of opportunity from B and C
-----  opportunity from each.  --A--  separately so the movement doesn't
--o--                          --o--  provoke again from B and C.

Fig. 5
-----  A exits B's and C's
-----  threatened areas as
-B-C-  part of A's move
-----  action, still not
--A--  provoking again
       because of movement.

Using the Skill Acrobatics
To avoid the attack of opportunity for leaving a square that B's threatening during Figure 3 A makes an Acrobatics skill check to Move Through a Threatened Area.

• Failure means A provokes an attack of opportunity from B. A continues taking his move action, provoking no further attacks of opportunity for normal movement from B.
• Success means A doesn't provoke an attack of opportunity from B but A's normal movement is halved while A continues his move action in B's threatened area.

To avoid the attack of opportunity for leaving a square that C's threatening during Figure 3 A makes an Acrobatics skill check to Move Through a Threatened Area. This Acrobatics skill check DC is increased by 2 if A has already succeeded on an Acrobatics skill check to Move Through a Threatened Area of another foe--y'know, like B--this round (see Table: Acrobatics DC's to Move Through Threatened Areas).

• Failure means A provokes an attack of opportunity from C. A continues taking his move action, provoking no further attacks of opportunity for normal movement from C.
• Success means A doesn't provoke an attack of opportunity from C but A's normal movement is halved while A continues his move action in C's threatened area.

Note 1: If there's a choice, the player chooses the order in which to make the Acrobatics skill checks versus the foes.

Note 2: I am of the opinion that movement isn't increasingly reduced by further successful Acrobatics skill checks as a cumulative movement penalty would be the result of a single source, but this is not stated by the Pathfinder rules.

References

• The skill Acrobatics' use Moving Through Threatened Squares. This really important sentence is the extent of the description given to the Acrobatics skill use Move Through an Enemy's Space: "If you attempt to move through an enemy’s space and fail the check, you lose the move action and provoke an attack of opportunity."
• The section of Movement, Position, and Distance detailing Moving Through a Square.
• The section of Combat Statistics detailing Attacks of Opportunity. This really important sentence is buried under the heading Combat Reflexes and Additional Attacks of Opportunity: "Moving out of more than one square threatened by the same opponent in the same round doesn't count as more than one opportunity for that opponent."
• The FAQ's entry for Acrobatics: How does Acrobatics work when you use it to avoid attacks of opportunity? When do you make checks? How many do you make?