The open position (the position of the hand when it is near the nut [the notched piece at the headstock-end of the fretboard]) is the most basic position, and has the slight advantage of easily employing open strings (strings played without being fretted).
In the diagrams on this site, the horizontal lines are the strings (the thinnest string is at the top of the diagram, as it appears to you when you tilt your head down and look at your own fretboard) and the vertical lines are the frets, except for the wider vertical line on the left which is the nut.
A hollow dot just to the left of the nut indicates that that string is played.
A solid dot on the string indicates that a finger is placed there. (To get the string to make good contact with the fret and avoid buzzing, you may need to place the fingertip just to the left of the fret.)
The numbers one through four on the dots indicate which finger is to be placed there, as follows:
The index or pointer finger
The middle finger
The ring finger
The pinky
A vertical bar spanning a few strings indicates that a single finger should be pressed flat to the fretboard, pressing multiple strings down. (This is called a barre, and is usually done with the index finger.)
An X indicates that this string is muted by letting one of your fingers touch it (usually a finger that's fretting a neighboring string at the same fret as the X appears).
Exercises
Play all the Major Triads, ascending and descending through the Chromatic Scale (C, C#, D, Eb, E, F, F#, G, Ab, A, Bb, B, C...), and the Cycle of Fifths (C, G, D, A, E, B, F#, C#, Ab, Eb, Bb, F, C). Repeat with the Minor Triads. Once you get this down, it will be very useful to begin doing your exercises with a metronome. You don't even need to buy one: There's a free one online at metronomeonline.com. The point is not necessarily becoming very fast--Keeping and updating a Metronome Exercise Record will simply provide a standard of both comparison between skills and measurement of progress. You will become able to analyze and concentrate on weak areas.