First Method

I will use this image to demonstrate. You may download it to practice with if you wish.

Open your image in your PSP workspace.

Take your Dropper tool    and right click on the background of your image.
This will place the background color of your image in the background color display of your color palette.  

Go to your Menu Bar, choose Selections, Select All.  
This will place a marquee around the entire image.

Go back to Selections and choose Modify. This will open a side bar; click on Transparent Color.

A window will open called Remove Selected Color. Choose Background Color from the drop down menu. The Tolerance should be at 1. Click OK.

Go to Color on the Menu Bar and choose Set Palette Transparency.

Choose Set the transparency value to the current background color.

You can see the background color in the area to the right. Click OK.

Now you will want to check to see what will be transparent in your image.

Got to Colors, Click on View Palette Transparency. Now you will see just what you have chosen to be transparent in our image. There will be a checkerboard pattern behind it showing the transparent section.

Save your image in gif format into the directory you choose.

Here is the image that I saved.  

As you can see, the white background has been rendered transparent.

I will use a different image to demonstrate this method. This is a little straw hat I made. You may download it for practice.  

This method uses the Export feature of PSP. There are many choices inside this feature. We are only going to examine the simplest way to save a basic Gif file.

First move the image you want to save to the far right of the work space.

Go to File, Export, Transparent GIF.

Click on Transparent GIF and the Transparent GIF Saver window will open.

The choice we want to make transparent is: Areas that match this color.

The color we want inside the box is the background color of the image, in this case the lavender.
How can we tell exactly what shade it is? Easy!

Move your cursor over to the image of the straw hat--it changes to a dropper tool! ( the image you moved to the right of the work space, NOT the image in the view window)

Just click on the background of the image and Presto! that color is inside the box.

At the same time, in the right preview window, you will see the effect this has; all the areas that color have turned transparent and have the checkerboard pattern to show it.

You can zoom in to see the image in more detail.
You can also see the reduction in file size because of using fewer colors. This is the Compressed file size.

There are many other options to choose from for different types of files, colors and images. This is the simplest method and should get you started saving your images to use on the web.

Click OK and the Save window will open to allow you to choose in which file to save your new transparent image.