There are a few widgets that are only widgets, like Beautiful Widgets. These are typically clock/weather widgets. For the most part, a widget is an extra feature that comes with some apps, and you can put the app, the widget, or both on you home screens. For example, you might have a calendar app that also has a widget that is always showing your upcoming appointments for the current day (thus keeping you from having to open the calendar app). As far as resizing, it probably depends on the widget and the amount of available screen size you have on a particular home screen.
Having said that, I'd caution against going widget crazy on your home screens. Personally, I've found many widgets to be serious battery drainers...because they are always updating data, as opposed to most apps that update when launched. Some calendar widgets I've tried have been really bad about this, but that doesn't mean they all are. Try some out. If you find your battery dying extra fast, get rid of them. I like my clock/weather widget, but other than that, I never considered it an inconvenience to tap an app icon.