Let me start by saying that if you don't know what scales and chords are, you may have a difficult time understanding the theory behind them. But I will do my best to make this short, sweet, and easy to understand:
A scale is simply a collection of notes/pitches that are used to create a piece. These notes can be limited by a repeating octave note, but they don't have to. For a more in-depth definition go here: http://en.wikipedia.org/wiki/Scale_(music)
Chords are simply any group of notes (3 or more) usually played at the same time to create harmony. Most chords relate to the scale you are using (the pitches of the chords are contained within the scale) Here's the official defintiion: http://en.wikipedia.org/wiki/Chord_(music)
Finger positions, aka, fingerings, are the patterns that are used to play scales, chords, arpeggios/broken chords, etc. They usually assign numbers to certain fingers (the numbers differ depending on the instrument...in piano: thumb=1, index=2, middle=3, ring=4, pinky=5)
Now for more theoretical info:
Here are the notes of a C Major scale to use as an example (playing this on the piano would really help to visualize as well):
C D E F G A B C
In theory, we use what is called Roman Numeral Analysis to help identify harmonic relationships in a piece of music. Do you know about tertian chords? They are chords made up of stacked thirds. I guess I didn't even think to ask...you do know what thirds are, correct? It will be hard to explain this without you knowing about intervals. This might help a little with intervals http://library.thinkquest.org/15413/theory/intervals.htm
http://en.wikipedia.org/wiki/Interval_(music)
Anyway...tonic and dominant does not refer to notes as much as they refer to chords or stacked thirds that are built from a specific note in a scale.
The tonic triad, is a chord that is built on the first (or last) note of a scale. In the case of C major, the tonic triad would be -C-E-G (a C major chord)
In roman numerals this is represented by a Roman numeral - I.
Dominant - is built on the 5th scale degree....in this case G, the chord would be G-B-D.
This is a very loose (and rather poor) explanation of this because there is so much you would need to know before really understand this... I would suggest reading up on basic theory: http://www.aboutmusictheory.com/harmonic-analysis.html
http://en.wikipedia.org/wiki/Diatonic_functionality
You can do searches for roman numeral analysis or harmonic analysis.
Cadences are a more complex idea...you would need to understand scales, chords, and tonic-dominant relationships before this. But for the sake of it...here is a general defintiion: A cadence is particular group of chords that ends a phrase...some chords/cadences make things sound more final than others. Here's a definition:http://en.wikipedia.org/wiki/Cadence_%28music%29
I hope this gets you started.