This is looking at the stakes of a game going into the game and is independent of how the game turned out (win or lose or future importance).  

For Utah (in chronological order)

1. 1994 #18 Utah vs #12 Colorado St: What I believe to be Utah's first ranked-on-ranked matchup, a matchup of undefeated teams and one of the best Utah seasons in a LONG time.  Utah win
2. 1994 Freedom Bowl #14 Utah vs #15 Arizona: Utah's first ranked-on ranked bowl game and (to this point) the lowest combined ranks in a game and secured Utah's first AP ranked season. Utah win.
3. 2003 Liberty Bowl #25 Utah vs unranked 9-3 Southern Miss: Another ranked vs unranked, but this game allowed Utah to finish AP ranked for only the 2nd time ever and first since 1994. Utah win
4. 2004 #5 Utah vs unranked 5-5 BYU: Weaker opponent, but a rivalry game where a win would punch Utah's ticket to be the first BCS buster.
5. 2004 Fiesta Bowl #5 Utah vs #19 Pitt: Pitt was the weakest BCS team, but Utah was still seen as a paper tiger, so had a lot to prove.  Ended up a huge Utah win, not even close.
6. 2008 #10 Utah vs #11 TCU: Utah had gotten to a high ranking playing mostly weaker teams, so #11 TCU was considered a true test and a huge game.  Utah win
7. 2008 #8 Utah vs #16 BYU: Rivalry game and Utah needed a win against a ranked BYU team to punch their ticket to a second BCS bowl.  Utah win
8. 2008 Sugar Bowl #7 Utah vs #4 Alabama: Alabama had spent the last several weeks at #1 before being upset by Florida in the SEC title game and dropping to #4.  By far Utah's biggest opponent to this point. Utah win
9. 2009 #18 Utah vs unranked 1-1 Oregon: Coming off their upset of Alabama, 2009 saw Utah start ranked.  They had a chance to prove they were for real against Oregon, who was a Top 10 team the previous season and had started ranked #14 before a surprise loss to Boise St.  Utah lost
10. 2009 #16 Utah vs #4 TCU: After the loss to Oregon, Utah worked their way back up and had another chance to prove themselves vs #4 TCU.  Utah lost (and lost badly)
11. 2010 #6 Utah vs #4 TCU: Both Utah and TCU were undefeated, heading into a huge clash in their last years in the MWC and both had eyes on not only getting into the BCS but possibly playing for a national title. Utah lost badly. (TCU went on to finish #2)
12. 2014 #19 Utah vs #20 USC: Utah's first ranked-on-ranked matchup in the PAC12 and a chance to prove they'd found their footing, going up against the marquee brand in the conference.  Utah win
13. 2015 #18 Utah vs #13 Oregon: Coming off their first ranked PAC season, Utah matched up against the best team in the PAC with a chance to make some noise. Utah win 
14. 2015 #3 Utah vs unranked 3-3 USC: This was the highest ranking Utah had ever taken into a game and they were matching up against the preseason #8 team.  At this point, the path to playoffs was wide open.  Utah lost.
15. 2016 #17 Utah vs #4 Washington: Just a big game with 7-1 Utah vs undefeated #4 Washington.  Utah loss
16. 2018 PAC12 Championship #17 Utah vs #11 Washington: ranked-on-ranked championship game for Rose Bowl berth
17. 2019 PAC12 Championship #5 Utah vs #13 Oregon: Utah was basically playing for a spot into the CFP (assuming Georgia would lose the SEC championship to LSU...which they did). A win puts them into the CFP, a loss would knock them into the Alamo Bowl.  Utah lost.
18. 2021 #23 Utah vs #3 Oregon: After a rough start to the season, Utah had reemerged in the rankings and was going up against a top title contender. Utah win.
19. 2021 PAC12 Championship #17 Utah vs #10 Oregon: Another matchup vs Oregon, this one for their first PAC12 title and a Rose Bowl berth.  Utah win
20. 2021 Rose Bowl #11 Utah vs #6 Ohio St: An NY6 bowl with huge following/viewership and a chance to make a major statement. Utah loss.
21. 2022 #7 Utah vs unranked Florida: Utah had been mentioned as a dark horse CFP candidate and opened against a traditionally strong (but unranked and coming off a bad season) SEC power in Florida. Big game that Utah needed to win. Utah loss
22. 2022 #11 Utah vs #18 UCLA: Voters had largely minimized Utah's very close loss to Florida and they were still in CFP contention, but faced a stiff test against a ranked UCLA team. Utah loss.
23. 2022 #20 Utah vs #7 USC: Coming off the UCLA loss and matched up against an undefeated CFP contender in USC. Utah win
24. 2022 #10 Utah vs #12 Oregon: Anytime you're a top 10 team playing a highly ranked team for a (near-guaranteed, so long as you beat Colorado) bid to the conference championship, it's a huge game. Utah loss
25. 2022 #11 Utah vs #4 USC: Playing a top-5 opponent for a spot in the Rose Bowl, otherwise will likely end up in Alamo, Holiday or Sun Bowl.  Certainly a big game, results TBD

Amazing how many of Utah's biggest games in history have happened in the last 5 or so years. Fully half of them have been since 2015.

Also a shoutout to 

1964 Liberty Bowl unranked 8-2 Utah vs unranked 7-3 West Virginia: Both teams would have likely been ranked, except that the AP poll at the time only ranked the Top 10.  But Utah was #14 in the pre-bowl coaches poll. Considering the losses by Oregon St and Syracuse and that #9, #10, #13 didn't play in bowls, Utah would have likely been borderline top 10 after the bowl games. Utah win

Probably should have included it, but I'm frankly not sure how to assess games from that far back.  There were also a couple undefeated seasons in the 1920s and I just basically ignored that far back.