It is generally thought that sushi originated in certain parts of Southeast Asia as a means of
preserving fish. Salted fish was packed with rice and allowed to ferment; when done, the rice was
thrown away and the fish eaten. Variations of this practice still exist, for example Kassam in
Borneo. This form of preservation travelled northward through China and into Japan.
Nare sushi is considered to be the direct precursor of modern sushi and is currently represented by Funa sushi. Nare sushi incorporates a stone weight to press the salted fish and rice and fermentation could last several years. Pressure is the innovation that distinguishes Nare sushi and modern sushi from the fermentation practices in China and Southest Asia.
Nama Nare sushi is essentially Nare sushi with a shorter fermentation time, about one month. In this form, the fish is slightly fermented and the rice edibly sour. This is the first form of sushi where both fish and rice are eaten much like modern sushi.
Haya sushi distinguishes itself from its ancestors by using vinegar to shorten the fermentation time even further. Even today, properly prepared sushi rice will have a faint whiff of vinegar. Haya sushi continues the use of a stone weight for pressure but fermentation is reduced to one day. Kiri sushi (and its modern descendant Hako (box) sushi) and Sasa Maki sushi are variations of Haya sushi. Another variation is Saba (mackerel) sushi which originally used a stone weight but now a bamboo roller.
The next innovation was pressure without a stone weight epitomized by Nori Maki sushi and Nigiri sushi. Maki is the generic name for any rolled sushi. Nori Maki is rolled sushi wrapped with dried seaweed and the pressure comes from using a bamboo roller. Nigiri sushi (also called Edomae sushi) uses hand pressure and is synonymous with modern day sushi. Another form of rolled sushi is Gunkan Maki, usually classified as Nigiri; rice individually wrapped and hand pressed with dry seaweed then topped (for example Ikura).
Sushi without pressure? Yes, it does exist in the form of Chirashi sushi, a bowl of rice laden with various toppings. Variants include Bara sushi and Mushi (steamed) sushi.
All sushi can be classified as one of the above but there may be other names used depending on local custom. Nigiri and Maki are what most westerners know and love; however you should not miss the exquisite taste and style of modern variants of Nama Nare and Haya sushi.