What do people mean by the "beam" profile/model pertaining to WMAP? Search results are all rather packed with jargon. Is there a pictorial /layman explanation of it and how does it affect the data?
The "beam profile" is what a point source would look like when observed by the instrument.
In general the observed spatial distribution or profile of any source in a WMAP map will be the intrinsic profile or distribution of the source convolved with the beam profile.
In layman's terms it is the amount of smoothing or blurring introduced into the WMAP pictures by limitations of the spatial resolution of the instrument.
The beam profile of a radio telescope characterizes its sensitivity to signals as a function of angle off axis. This is usually divided into a main beam in the target direction and sidelobes in other directions. As this figure from Hill et al. 2009 shows, the WMAP main beams and near sidelobes were rather complex:
Image credit: WMAP Science Team. Each square covers 10x10 degrees. Top row: beam maps based on multiple observations of Jupiter. Middle row: beam models fitted to the observed maps. Bottom row: residual differences between top and middle. Left and right columns: dishes A and B, which point in different directions.
Without further processing, the image of a point source resembles the beam profile. To reconstruct a clear image, astronomers can deconvolve the observed data with the beam profile. Lest this operation amplify noise from the observed beam map, they use a smooth mathematical model of it instead. The WMAP team developed their beam model using physical optics software called DADRA (Diffraction Analysis of a Dual Reflector Antenna).
The far sidelobes were especially important for WMAP. The sky has several radio sources, e.g. our galactic center, much stronger than the desired background signal. To minimize foreground contamination, the WMAP team made an all-sky sidelobe map based on multiple observations of the Moon. Then they made a mask of the strong sources, convolved it with the sidelobe map, and subtracted the result from the observed signal.
For details of how and why they did all that, see the "Ancillary Data" chapter of the WMAP 9-year Explanatory Supplement and the other publications cited there.