Resolution is a measurement of how much information, as represented by dots or pixels per inch (DPI/PPI), is contained in a given picture file. It's true, the higher the resolution, the larger the file size. But, the higher resolution, the more you can do with that picture.
Pictures intended for print require a resolution of at least 300 DPI or higher, whereas pictures destined for the web are usable at 72 DPI because that's the highest resolution monitors/screens can display.
You do not need professional photography equipment to take high-resolution pictures - A point and shoot camera will do just fine.
Most digital cameras take high-resolution pictures. However, because high-resolution pictures capture more visual information (pixels), the file sizes are larger and tend to fill up a memory/data card more quickly. If all you ever intend to do with your pictures is share them electronically, then low-resolution settings will suffice because the image never leaves the screen/computer/72 DPI environment.
However, if you think you might like to print your pictures one day…
The resolution solution:
Make sure your digital camera settings are configured to take pictures at the highest resolution possible. Having high-resolution source files will enable you to use your pictures for more multiple purposes - print, web, etc.
Pictures taken at a high resolution can be resized/resampled/optimized for the web, but the reverse is not true. Pictures taken at a low resolution cannot be resized/resampled/optimized for print.
Worried about running out of room on your memory/data card? Buy a larger one or a back-up card. It's always good to have a spare.
Download your pictures to your computer on a regular basis. There's nothing worse than losing a whole year of pictures and memories because you never backed them up.