Title tags may be used with links. The title tag displays when you mouse over the link, thus providing more information about the link. You may use the title tag to let the person know what is coming next; for example, "click here will take you to our website" would display on mouse over. This does not mean you should not use your keywords with your link.
Both alt and title tags are visible to the search engines. Search engines need the alt tag to see an image. Search engines use the alt tag and the graphic name, i.e., mydog.gif, to categorize images.
Do not, I repeat, do not use the same keywords in your title or alt tags. Use only what you need to help people navigate your site. Keyword stuffing is black hat SEO and will get you banned from search engines. Alt and title tags are all about accessibility, with a nod given to SEO.