Patterns, tiles, characters - small pieces of graphics that are 8x8 pixels. NES can handle two sets of 256 characters. They are used for backgrounds and sprites. Only one set can be assigned to background or sprites, i.e. you can't use characters from different sets on a single screen (without tricks), but you can use one set for background and other for sprites.
Sprites - small movable objects that are overlayed on the background and can move independently. Graphically one sprite represented with one character 8x8 pixels, taken from one of the character sets. Up to 64 sprites could be displayed on a screen, but only 8 on a scanline (one horizontal line of pixels).
Background - an array of 32x28 cells. Sort of a text screen, one byte per cell - a number of a character in bank. It could be scrolled by two axises, but the cells always keep their relation to each other.
Nametable - part of video RAM that forms the background, 32x28 bytes. There are two nametables (technically four, but console only has enough video RAM for two), to organize a bit larger scroll area, also useful for some tricks.
Tricky part of NES graphics is the color.
There is general palette of 56 colors, defined by the hardware design, and a programmable palette of 32 entries. Each entry could have an assigned color. The entires are divided into two halves - 16 entires for background graphics, 16 for sprites. These further divided by four parts by 4 colors each, these are called palettes too. One sprite can have one of the four sprite palettes assigned. For background it is more complex, palettes assigned to blocks of 2x2 cells rather than individual cells.
Despite there are 32 palette entries, only 25 colors could be displayed at once (without tricks). That's because first color of all four background palettes is always the same, the background color - so 1+3+3+3+3=13 colors for background. First color of a sprite palette considered 'transparent', i.e. you can see background through it - so 3+3+3+3=12 colors for sprites. 13+12=25 colors on the screen at once.
There are many little details as well, ask if you want to know something specific.
You can try and see all these NES graphics components and their relation in action using my NES Screen Tool - a specialized nametable editor that also allow to create graphics.