You can upload some of your albums to the web. Go to Picasa Web, get an invitation for the limited beta, download the new version of Picasa (even if you already have Picasa, you need to upgrade). When you right-clik an album, you'll see a new option: "Upload to web album". You have a space limit of 250 Mb and you can make a web album public or private. For $25 a year, you can have 6GB extra storage space and no bandwidth limits.
For an album, you can specify a name, a description and a cover album. You can add captions by clicking "Add Caption" while viewing individual photos.
You can also use Picasa Web Albums without installing Picasa using a basic uploader or a more fancy ActiveX control that allows you to upload more pictures at once (Internet Explorer). To do that, visit Picasa Web and click "New Album" or select the album you want to upload photos. Click "Upload Photos" on the left side of the page.
By default, the URL of your public web album is based on your Gmail username, but you can change that in your settings.
You can also download other users' public albums to Picasa, so this is a great way to share photos.
Unfortunately, it's very hard to organize large albums on the web. There is no search option, you can't assign tags and there is no way you can synchronize an album from Picasa to a web album.