The yellow umbrella was a cornerstone of How I Met Your Mother, and it make several appearances throughout the show's run. The yellow umbrella served as a significant object throughout the sitcom since it connected Ted Mosby and the "Mother" years before they officially met. If it wasn't for that yellow umbrella, Ted may have never walked up to his future wife (and, in turn, ultimately had his kids to tell this story to).
Before the Mother was officially introduced in How I Met Your Mother, she was often featured hidden by her identifiable yellow umbrella. She was the original owner of the umbrella but it came into Ted's possession for a brief time before it went back to the Mother. When Ted was at the Farhampton train station after Barney and Robin's wedding, he saw the Mother holding the yellow umbrella. He walked up to her and claimed that it was his umbrella and it had his initials, TM, on it. The Mother then introduced herself as Tracy McConnell and explained that she bought the yellow umbrella but lost it for a few years. Ted then filled her in on how he got a hold of it, and that instant connection would turn into a romantic relationship.
The yellow umbrella was a symbol of Ted and Tracy's relationship even before they directly met. In order to keep that connection in How I Met Your Mother, the umbrella would appear from time to time across the show's nine seasons. Its first appearance came in the season 3 premiere, "Wait For It". Ted was still upset about his breakup with Robin, but one of the final scenes of the episode showed someone walking past MacLaren's Pub using a yellow umbrella. This was the earliest indication that the Mother was out there, but Ted still had a journey ahead of him before they met. The yellow umbrella then made an appearance or was referenced at least once in every season that followed.
The story explaining how Ted got the yellow umbrella was revealed in season 3, episode 12, "No Tomorrow". Ted went to a St. Patrick's Day party at a club with Barney (and notably encountered the Bump Girl) and ended up leaving his phone there. He returned the next morning to retrieve his phone but it was raining, so he grabbed a forgotten yellow umbrella in the club. That umbrella was left there by Tracy who also attended the party. Ted later used the umbrella again when he encountered Stella in the season 4 episode, "Right Place, Right Time". The meeting would lead to his teaching job at Columbia University which is where Tracy saw Ted for the first time. Then in season 5, episode 12, "Girls vs. Suits", Ted went on a date with his student, Cindy. He arrived at her apartment with the umbrella and left it by her door but forgot it when he left. It turned out that Cindy was Tracy's roommate so the yellow umbrella returned to its original owner.
In How I Met Your Mother season 6, a flash-forward scene in "Big Day" showed Ted at a wedding (later revealed to be Barney and Robin's) when he realized he forgot to bring an umbrella. This was foreshadowing that the umbrella would come into play later that weekend. The following season, Ted found closure with Robin, and in the episode "No Pressure", he walked outside to find everyone using a yellow umbrella, indicating that his future was still bright.
The series first teased Ted and Tracy's meeting in the season 8 premiere, "Farhampton", when the Mother was shown at the train station, standing near Ted. Later that season, during the episode "Something New", the Mother's face was finally shown as she purchased a train ticket to Barney and Robin's wedding, where she would be playing in the band. At the time, she was holding her guitar and the yellow umbrella. Aside from the How I Met Your Mother series finale, the yellow umbrella made one more appearance in the season 9 episode "How Your Mother Met Me". The episode was told from Tracy's perspective and explained all of the chance encounters she had with Ted by tying together the events of the significant umbrella.