Blogger is integrated with other Google services, which makes adding videos to your blog posts pretty easy. There's a video button at the top of the "Create a new post" form. That opens a selection box with multiple options. You can search YouTube for a video to embed in your post, or add a video of your own that you've already uploaded to YouTube. Alternately, you can upload a video directly from your computer – this video will be hosted by Google Video, and isn't searchable or publically available, unlike videos uploaded to YouTube (it will still show up in your blog post if you embed it there).
Finally, there are options for uploading video from your Android phone (which requires a free app) or from a webcam.