Offensive Strategies
The first step in eliminating your opponent's ships in a game of Battleship is to find them. In order to do this, it is best to cover a lot of space quickly. Large sweeping patterns will therefore be useful. However, it is important not to make a sweep too large. For example, if you test every fifth square along a row, you are likely to miss smaller ships.
A smaller sweep, testing every other square in a row, will be able to locate every ship. However, this sweep will take far too long; your opponent might find all of your ships before you find even one of hers!
Use a searching pattern that involves medium spacing, testing every three squares. This will occasionally miss the two-unit ship, but you will be able to find most of your opponent's ships quickly.
Once you score a "hit," you should sink that ship. On your next turn, abandon your sweeping pattern and test one of the immediately adjacent squares. Continue until you have sunken the ship, and then resume your sweep pattern.
It is important to consider the size of ships you have sunk in relation to the size of undiscovered ships. Vary the size of your sweep accordingly; if you sunk your opponent's 5-unit ship, narrow your sweep spaces.
It is also important to consider the psychology of your opponent when beginning a sweep. If she assumes you will start your search in the upper left of the grid, perhaps she placed her ships elsewhere. For this reason, you should also move randomly within your sweep pattern; don't just move from right to left or top to bottom.
Ship Placement
It is best to place your own ships as "randomly" as possible, so your opponent has difficulty predicting where they lie.
Clustered ships can be an effective strategy, as your opponent might not expect it. Additionally, if you cluster your ships, your opponent will probably have difficulty determining which of your ships she has sunk. Furthermore, if she often resumes her search in a distant corner of the grid, it will take her longer to find your ships if they are close together.
If you and your opponent know each other, use that psychology to your benefit. If she thinks you always place your ships close together, opt for a more spacious layout. On the other hand, if you often place your ships far apart, place them close together occasionally.
Ultimately, psychology is the most significant aspect to consider when placing your ships. Place your ships where your opponent least expects them.