Cursor movement

w	jump by start of words
W	jump by words (no punct.)
e	jump to end of words
E	jump to end of words (no punct.)
b	jump backwards by words
B	jump backwards by words (no punct.)
0	start of line
^	first non-blank character of line
$	end of line
G	GoTo command (prefix like 5G)

Note:
Prefix a cursor movement command with a number:
for example, 4j moves down 4 lines.

Insert Mode - Inserting/Appending text

i	start insert mode at cursor
I	insert at the beginning of line
a	append after cursor
A	append at the end of line
o	open (append) blank line below current one
O	open blank line above current one
ea	append at the end of word
Esc	exit insert mode

Editing

r	replace a single character
J	join the line below to current one
cc	change (replace) an entire line
cw	change (replace) to the end of word
c$	change (replace) to the end of line
s	delete character at cursor and substitute text
S	delete line at cursor and substitute text
xp	transpose two letters (delete+paste, technically)
u	undo
ctrl+r	redo
.	repeat last command

Marking text (visual mode)

v	start visual mode
V	start Linewise vidual mode
o	move to other end of marked area
ctrl+v	start visual block mode
O	move to other corner of block
aw	mark a word
ab	mark () block (incl. braces)
aB	mark {} block (incl. brackets)
ib	inner () block
iB	inner {} block
Esc	exit visual mode

Visual commands

>	shift right
<	shift left
y	yank (copy) marked text
d	delete marked text
~	switch case

Cut and Paste

yy	yank (copy) a line
2yy	yank 2 lines
yw	yank word
y$	yank to end of line
p	put (paste) the clipboard after cursor
P	put (paste) before cursor
dd	delete (cut) a line
dw	delete (cut) the current word
x	delete (cut) current character

Exiting

:w	write (save) the file, but don't quit
:wq	write (save) and quit
:q	quit (fails is anything has changed)
:q!	quit and throw away changes

Search/Replace

/pattern	search for pattern
?pattern	search backwards for pattern
n	repeat search in same direction
N	repeast search in opposite direction
:%s/old/new/g	replace all old with new throughout file
%s/old/new/gc	... with confirmations

Working with multiple files

:e filename	edit a file in a new buffer
:bnext (or :bn)	go to next buffer
:bprev (or :bp)	go to previous buffer
:bd	delete a buffer (close a file)
:sp filename	open a filename in a new buffer and split window
ctrl+ws	split windows
ctrl+ww	switch between windows
ctrl+wq	quit a window
ctrl+wv	split windows vertically