In How I Met Your Mother, the yellow umbrella is a symbol of hope. It represents the idea that no matter what happens in life, there is always a chance for happiness and love. The yellow umbrella first appears in the show's pilot episode, when Ted Mosby (Josh Radnor) sees it floating through the air and decides to chase after it. The umbrella eventually leads him to meet his future wife, Robin Scherbatsky (Cobie Smulders). Throughout the series, the yellow umbrella pops up at various important moments in Ted and Robin's relationship, serving as a reminder that they are meant to be together.
The most significant appearance of the yellow umbrella comes in the season 8 episode Something Old. In this episode, Ted is getting ready to marry his girlfriend Victoria (Ashley Williams), but he starts having doubts about whether she is really The One. He begins to think back on all of the special moments he shared with Robin throughout their relationship, including those times when they were saved by the yellow umbrella. As he does so, he realizes that he still loves Robin and that she is his true soulmate. He ends up leaving Victoria at the altar and running off to pursue a relationship with Robin once again.
While Ted and Robin don't end up together in the end (she marries someone else), their story is still one of hope and enduring love. The fact that they are able to stay friends despite everything they've been through is a testament to how strong their bond really is. And even though Ted doesn't get his happy ending with Robin, he does find happiness with someone else - which wouldn't have been possible without that little yellow umbrella bringing them together in the first place.