Not sure why that's happening to you. It looks like the component installed correctly. Try just installing the plugin. Here is how:
1. Unzip the extension install package, i.e. Joomla_1.5_pkg_nicetweets_v2.25-full.zip
2. Find the unpacked file at com_nicetweets/admin/extensions/plg_nicearticletweets_v2.03.zip
3. Use Joomla's standard extension installer to install plg_nicearticletweets_v2.03.zip.
That should do the trick. Let me know if it doesn't.
~ Best regards