Lisa, the objective is to place the numbers in such a way so each horizontal row, each vertical line and each smaller box with 9 squares contain all the numbers from 1 - 9 with no duplications.
Yes, it's random, in that as a rule you can't do it in numerical order. It's also not mathematical, it's pure logic. And the numbers already placed there - they are to give you a start. Generally each puzzle has one, unique, solution.
So look at the puzzle - it has nine smaller 9 square boxes in it. It also has 9 vertical lines and nine horizontal rows. You need to place the numbers so they fulfil the requirement of no doubling up, no leaving a number out.
The first and most important guide - DO NOT GUESS. This is pure logic. You only place a number when you are SURE that it goes there. If you have a choice of two digits and you flip a coin, it means you have a 50:50 chance that you have just totally SNAFUd the puzzle.
Sometimes the position of a number will make it easier - it removes one possible position and also gives you a guide. This includes the numbers that YOU put in, certain that you have put it in correctly.
Example - look for the same digit in three consecutive either vertical rows or horizontal lines. These should then also cover three boxes (the 9 square boxes). A particular number in vertical line 1 (on the left) in the top box means that for all boxes below that top box, that number cannot be in line 1. If, in the next box down, you have the same number in line 2, then you KNOW that in the bottom box under the first two, that same number HAS to be in line 3.
So then you look at the rows - does the same number appear in any rows that intersect with the boxes in question? Look especially at the bottom three rows - these can help you place the number, if possible, in the bottom left box. If these help you place the number exactly, that's one more piece of the puzzle done.
If you're not sure, and especially if you're just beginning or it's a tricky puzzle, you can lightly pencil in to one corner the possible numbers for that placement. Then when you are sure, you can go back and write your certain answer in.
Do the same for other positions and other numbers. Over and over - often you can't pin a position down early on, but after you've placed a few numbers you go back and test it again, and find that your placement has eliminated one of two possible placements, leaving you with certainty - gold, in this game.
For those who are more practised, you can apply the same principle with pairs of possible numbers - if you see a 1 and 4, for example, in the top box and both in line 1, and you see a 1 and 4 in the second box in line 2, then you KNOW that 1 and 4 MUST be in the bottom box in the third line.
It's a really good game for logic-minded difficult children, too - start them off on the easy puzzles and maybe work on a puzzle together until you both get the knack. Then you can print off puzzles and race each other to a conclusion, or simply enjoy time together, puzzling.
I find Sudoku a really good brain-stretching activity which you can also put down immediately you need to do something else - great for parents. I keep a book in the car, beside the bed, in the bathroom - in my pocket, even. Plus I have Sudoku on my computer so if I'm at a rather boring meeting and I'm up to date with my notes, while people are discussing something irrelevant I'll Sudoku away until they need me again. It's an alternative to solitaire.
Here's a link to one of my favourite websites - http://www.websudoku.com
If you're a beginner, start with the easy ones. If you play online it will give you hints or tell you if you've put a number in wrong. You can also print them out to work on at your leisure, but then the only way you will know you've got it, is if you finish the puzzle with all numbers placed according to the rules.
I do get a regular Sudoku book but it doesn't come out often enough for me and I've worked my way through the easy puzzles and like really tricky ones, these days. There are also some variations such as Samurai Sudoku, which can be scary if you're a beginner, although I think they're even easier because there are more clues.
One thing to always bear in mind with puzzles - they won't ask the impossible of you. There is always a path to a solution.
Maybe this is a good rule to remember when dealing with our kids, too!