I'll be quoting NFS@Home administrator with a few explanations about the goal of this Boinc Project.

The goal of NFS@Home is to factor large numbers using the Number Field Sieve algorithm. After setting up two polynomials and various parameters, the project participants "sieve" the polynomials to find values of two variables, called "relations," such that the values of both polynomials are completely factored. Each workunit finds a small number of relations, typically a bit less than 2,000, and returns them. Once these are returned, I combine all of the relations together into one large file then start the "postprocessing." The postprocessing involves combining primes from the relations to eliminate as many as possible, constructing a matrix from those remaining, solving this matrix, then performing square roots of the products of the relations indicated by the solutions to the matrix. The end result is the factors of the number. Currently, we are factoring about one number each week. All factors are linked from the status page.

For a (much) more technical description of the NFS, see the Wikipedia article or Briggs' Master's thesis.
My interest lies in the continued development of open source, publicly available tools for large integer factorization. Over the past couple of years, the capability of open source tools, in particular the lattice sieve of the GGNFS suite and the program msieve, have dramatically improved. My collaborators and I have factored quite a few large numbers using these tools.

Integer factorization is interesting both mathematical and practical perspectives. Mathematically, for instance, the calculation of multiplicative functions in number theory for a particular number require the factors of the number. Likewise, the integer factorization of particular numbers can aid in the proof that an associated number is prime. Practically, many public key algorithms, including the RSA algorithm, rely on the fact that the publicly available modulus cannot be factored. If it is factored, the private key can be easily calculated. Until quite recently, RSA-512, which uses a 512-bit modulus (155 digits), was used. As recently demonstrated by factoring the Texas Instruments calculator keys, these are no longer secure.

For most recent large factorizations, the work has been done primarily by large clusters at universities. There are two other public efforts, NFSNet and MersenneForum, in both of which I have participated, but the software used by NFSNet doesn't incorporate the latest developments and participation in the MersenneForum effort requires manual reservation and submission of work. I have been toying with the idea of trying a BOINC project for a while now to make it easy for the public to participate in state-of-the-art factorizations, and I found the time to do so. My interest in this project is to see how far we can push the envelope and perhaps become competitive with the larger university projects running on clusters, and perhaps even collaborating on a really large factorization.

The numbers are chosen from the Cunningham project. The project is named after Allan Joseph Champneys Cunningham, who published the first version of the factor tables together with Herbert J. Woodall in 1925. This project is one of the oldest continuously ongoing projects in computational number theory, and is currently maintained by Sam Wagstaff at Purdue University. The third edition of the book, published by the American Mathematical Society in 2002, is available as a free download. All results obtained since the publication of the third edition are available on the Cunningham project website.

Concerning target size, I started out somewhat "small" (small meaning that I can do it on our cluster in a few days), but it also took the contributors to this project only a few days as well. For our second project, I chose a target somewhat larger. I will continue to slowly increase the size. The real limit is the memory requirement. The current project requires a bit less than 512 MB of memory. As the size of the target increases, that will also increase somewhat. I will be keeping a close eye on that as we move forward.
One of the goals of the project is to collaborate with other groups on record-size factorizations.
Thanks to recent advances in the msieve postprocessing code, a significant barrier limiting the size of numbers that we can factor has been removed. Therefore, over the next few months, we will be factoring progressively larger numbers using the lasievef application. To reflect the increased importance of these larger work units, their credit has been adjusted.

If your computer has at least 1.25 GB of memory per core and you are not running other memory-hungry applications in the background, please try disabling the lasievee application in your NFS@Home preferences. If your computer can successfully run the lasievef application, you will contribute to the largest NFS factorizations completed to date using open source software and will earn more credits per day. In the case of a quad-core computer with 4 GB of memory, you will earn more credit per day running three instances of lasievef and leaving one core idle than running four instances of lasievee.

Do not worry if your computer does not have sufficient memory for the lasievef application or other background processes prevent you from using it. Work for lasievee will continue to be available over the long term.