Weak Two bids
Weak Twos are an extension of the traditional pre-emptive bids which were based on a seven card suit, and opened at the three level.   3 level pre-empts are very difficult to counter, but fortunately they do not crop up frequently.

Weak hands with six card suits are much more common. 
Previously these hands were passed and you had to wait for a chance to overcall. 
This meant that their pre-emptive value was lost. 

Variations on these two level pre-empts are:
1. A three way multi two diamond bid. 
2. Weak twos in diamonds, hearts & spades.
3. Weak twos in the majors. The most popular variation is Benjaminised Acol or Benji for short.

When to open a Weak Two
If you decide to make a weak two bid, it should fulfill all of the following criteria:-
1.) You hold a six card major.
2.) You do not hold four cards in the other major.
3.) You have  6-10 HCPs 
4.) You are no worse than an eight loser hand. or
5.) If you are a seven loser, you do not hold three controls for a one level opening.
6.) Your hand does not comply with the rule of 19 for one level opening. Known generally as 'Benji Acol' this system attempts to get the best out of playing weak twos while retaining all the advantages of playing strong twos. Indeed they now offer you the opportunity of showing a nine playing trick hand in clubs/diamonds which you couldn't do in traditional Acol.

Opening 2 level bids in Benji 
2C = An 'Acol two' in any suit. (or a balanced 21/22)
2D = 23+ points or game force. (equivalent to the 2§ in Acol).
2H = 6-10 points with a six card heart suit.
2S= 6-10 points with a six card spade suit.
2NT = 19/20 Balanced.

Responding to an opening bid of 2C by partner 
2C - 2D is, a forced relay bid. (No choice)

Opener's rebid
2H = eight playing tricks in hearts.
2S = eight playing tricks in spades.
2NT = 21/22 balanced 
3C = nine playing tricks in clubs.
3D = nine playing tricks in diamonds.

Responding to an opening bid of 2¨by partner 

2D -    2H is, negative 0-7 pts. (similar to 2C-2D in traditional Acol).
          2S/3/3C/3D shows 8+ and a five card suit. 
          2NT shows 8+ balanced. 

The auction proceeds as in traditional Acol.

Ogust responses

Originated by Mr. Harold Ogust, the Ogust convention allows the responder to make a forcing bid of 2 NT to allow opener, by a system of Relay bids, to describe his hand more precisely.  
Several variations on this concept are played.  e.g. After an opening bid of 2 - 2NT

3C :Shows minimum strength, poor suit
3D :Shows minimum strength, good suit
3H :Shows maximum strength, poor suit
3S :Shows maximum strength, good suit
3 NT:Signifies a solid suit