Difference between 'conventional' and 'adaptive' beamformers?
A beamformer is basically a spatial filter.  It can be passive, just like a temporal filter.  

Instead of samples separated by time, they are separated by space.  A passive temporal filter can be a bandpass that is "aimed" or "steered" at a particular frequency.  For passive spatial filters (i.e. beamformers), the filter can be steered towards a particular angle of arrival, instead of temporal frequency.

Adaptive filters/beamformers can be incredible because they can "steer" a null towards a frequency or angle where there is an interfering signal.

Just like a temporal adaptive filter, a spatial adaptive filter (i.e. beamformer) is constantly adjusting the filter weights/coefficients to optimize some criteria that usually involves "nulling out" or "rejecting" an interferer.

Here's a diagram of a CBF that is called a "K-Omega" Beamformer.


Here's a diagram to reinforce the idea that a passive beamformer is possible.



I realize that these are a bit random, but hopefully you can follow the logic of the above.
I'll see if I can find a better diagram that more clearly shows what's happening.  To be clear, the windows referred to above are Hamming/Hanning type windows, and this process is basically returning a 2-d matrix where temporal frequency is on the x-axis and a special spatial variable is on the y-axis.  This special variables makes the math easier, and it's a one step process to convert the special spatial variable and temporal frequency to an angle of arrival.

The green lines below are lines of constant angles.


Keep in mind that all of this is for CBF (conventional beamformers) and the above covers all spatial and temporal frequency (within the limits of Nyquist.
Some common ABF techniques are:

MPDR - Minimum Power Distortionless Rejection BF (Described by Van Trees)
DMR - Dominant Mode Rejection BF (Abraham and Owsley)
R-DMR - Robust Dominant Mode Rejection BF (Cox and Pitre)
EBAE - DMR BF with Eigenvector Beam Association and Excision (Kogon)


Instead of using a spatial FFT as the spatial filtering operation, these techniques typically involve constructing an updated spatial autocorrelation matrix of the incoming signal, and then using that matrix or eigenvectors of that matrix to adaptively influence the spatial filter.

Update for @Mohammad:
Below is the list that of beamforming texts I received from my professor:


Van Veen, B.D.; Buckley, K.M.; , "Beamforming: a versatile approach to spatial filtering," ASSP Magazine, IEEE , vol.5, no.2, pp.4-24, April 1988
Efficient digital beamforming in the frequency domain
Brian Maranda, J. Acoust. Soc. Am. 86, 1813 (1989)
Array Signal Processing by Don Johnson and Dan Dudgeon, Prentice Hall, 1993
Optimum Array Processing by Harry L. Van Trees, Wiley, 2002.


Also, after searching around, this looks very interesting.  It seems to be more interested in the practical/implementation side than the theory.  I don't have it but I'll probably buy a copy:


Practical Array Processing by Mark Sullivan, McGraw Hill, 2008 ( http://www.practicalarrayprocessing.com/index.html )