The term 'lymphoma' is nearly as generic a term for a disease as 'cancer'. The noun suffix '–oma' derives from the Greek and indicates a tumor, so if you imagine all the types of cancer like a family tree, at the very top you might have the term 'Cancer' and underneath it, like children, you would find Sarcoma and Carcinoma.
Sarcomas are cancers of connective tissue and although they overlap one another they tend to get their names from the most developed or mature tissue in the tumor. For example, an angiosarcoma suggests blood vessels, and an osteocarcoma suggests bone.
Carcinomas are cancers of surface (i.e. epithelial) tissue such as the lining of an organ. Renal cell carcinoma is cancer of the kidney, for instance. And all lung cancers, as well as skin cancers, are classified as carcinomas (melanoma is also known as melanocarcinoma). They might be called adenocarcinoma or squamos cell carcinoma, but they can be traced back to epithelial tissue.
These categorical terms were created when cancer wasn't well understood, and when scientists believed all cancers could fit nicely under these terms. Over the last several decades our understanding has grown enormously, and the terminology used to represent that understanding has not done a good job of remaining organized.
Lymphomas are cancers that develop in lymph cells found in the lymphatic system, either in tissue, nodes, or lymphatic organs. Lymphomas that develop from B-cells are therefore of B-cell origin and are B-cell lymphomas. Those that develop from T-cells are of T-cell origin and are T-cell lymphomas.
From there, the number of subtypes of B-cell and T-cell lymphomas has grown from ten to 30 to 60 and will very likely be in the high triple digits soon. As lymphomas become more and more classified at the molecular level, the list of subtypes expands exponentially.
While some of these new subtypes have names—follicular lymphoma, diffuse large B-cell lymphoma, anaplastic large cell lymphoma, angioimmunoblastic T-cell lymphoma—many do not; they are simply referred to by the chromosomal translocation that is distinctive to that tumor. You can get an idea of what I mean by visiting the classification of B-cell lymphomas at the Atlas of Genetics and Cytogenetics in Oncology and Haematology.
While there are important differences between a Hodgkin's lymphoma and a non-Hodgkin's lymphoma, you should keep in mind that Hodgkin's lymphomas are almost always lymphomas of B-cell origin. The three major differences are: 1) Hodgkin's lymphomas are characterized by the peculiar shape of the cancerous cell; 2) Hodgkin's lymphomas progress in a very orderly manner; 3) Hodgkin's lymphomas are far more common among people under the age of 30, whereas the opposite is true of non-Hodgkin's lymphomas.
There are many reasons why lymphoma patients should know which subtype they have, which is why the Leukemia & Lymphoma Society recently published the Know-Your-Subtype Brochure, a printable brochure that patients can have their doctors fill out that gives the specifics of their subtype of lymphoma. Knowing the subtype can help a patient research the subtype online, discuss it in forums, and even peruse the published treatment guidelines to see if the treatment they receive conforms to the standards being given at other medical centers.