These terms are all related to the SEO character of a page. Let's take them up one by one.
noindex - It works just like it sounds. It directs search engines to not index a given web page. The reasons can be many. Perhaps you want to discuss something which is private to your company and you don't want it to be listed publicly when people search about related keywords on the web.
doindex - This term is somewhat obsolete now and has no special effect or use and merely adds to the HTML of your webpage because even if you don't add it, search engines are likely to index the page anyway.
nofollow is a value which is set to the rel attribute to an HTML 'a' element and directs search engines to not follow a certain hyperlink mentioned in the webpage. This keyword is mainly used to ensure that too many outward-directing links don't hamper the PageRank of the websites. It is likely that if search engines are allowed to follow every outward link on a website, then the ranking of the website falls. This is the reason why many bloggers use this attribute for innocuous backlinking purposes.
dofollow is just the opposite of nofollow and allows search engines to follow the link present on a webpage. dofollow links are supposed to benefit the linked website in increasing their PageRank. One of the most famous and controversial ways of increasing the PR of a website is to have too many dofollow links to that website, which is basically known as backlinking.
Link Juice - This is a relatively less popular term, when compared with the rest discussed in here. Link juice is rather a colloquial term. It refers to the power that an external website passes on to your website via linking. The more the link juice, the better the search ranking of the website to which the link juice flows. This term is directly related to link-building strategies and determining the PageRank of a website. The more number of external links that follow to your website, the more link juice and the better the ranking.
Quote: The most dangerous person in this world is one who knows his strength but does not know how to terminate it if applied.