You want the short answer?
Combine any edible thing with a special type of cooked cooled vinegared rice and you have sushi. Take away the 'sushi-rice' and you have Sashimi.
While there are a few extra details, such as sashimi is always meat and always raw, and the rice is mostly a variety called awase-zu, this definition will generally hold true. But the complicating factor is that the range, the variety and the types of sushi vary significantly.
The two most common are:
O-Nigiri - a slice of raw (or cooked) fish (or meat) (and/or vegetable) on top (or inside) of an oblong (or square or triangle) ball (or other) shape of rice.
Maki - a strip of fish and/or vegetable rolled inside a piece of nori (seaweed) using a bamboo mat. This is most commonly known as a 'sushi roll'. You can also get 'temaki', a cone shape instead of a cylinder, or 'uramaki', an inside-out roll with the sushi rice as the outside layer.
Not to mention, Inari, a pouch of fried tofu filled with sushi rice, or Chirashi sushi, a bowl of sushi rice with the fish and other ingredients scattered throughout.
I love sushi, (perhaps not as much as Ramen), and I have tasted every type mentioned above. I have had the good, the bad and the unbelievable.
So what's my favourite?
In my humble opinion, if you have a good piece of fresh smooth well-cut fish, full of flavour, you can ditch the rice. In those circumstances, I will always say Sashimi wa, mo hitotsu kudasai, if not mo futatsu kudasai.
And just for fun, check out this fun and famous onigiri children's song. It doesn't help working out the difference between sushi, sashimi and o-nigiri, but is just plain cute.
Please note, O-nigiri is simply a polite way to say nigiri, and sushi often becomes 'zushi' when following a prefix, eg. maki-zushi, inari-zushi