1. Goldeneye
2. The World is Not Enough
3. Tomorrow Never Dies
4. Die Another Day
I do not think he made the worst films in the series (those honors go to Diamonds Are Forever, The Man with the Golden Gun, Octopussy, Moonraker and A View to a Kill).
It may skew my opinion as I grew up with Pierce Brosnan, but he is the second best James Bond after Sean Connery and he revived the series from the failure of the Dalton era as much as Craig brought the series back from the brink of absurdity (i.e. the second two acts of DAD). I think now that the "newness" of Daniel Craig's James Bond has worn off, many are starting to understand that. QOS was no homerun and IMO is as forgettable and mildly entertaining as Tomorrow Never Dies was (or many of the middling Bond films).
As to those who say they don't get why Goldeneye is so popular...
1) For older fans, as The Chairman pointed out, it saved the franchise. It was post-Cold War and Bond was considered a sexist relic (and he kind of is) that was obsolete. Brosnan combined the darkness of Connery with the humor of Moore and added his own 1990s "sensitive" side to Bond. It worked very well.
2) For younger generations (including myself) it was the first Bond to be seen in theaters. And unlike older installments, it wasn't creaky around the edges. It in short felt like a cutting edge action movie that had all the Bond charms and felt like the perfect product for a mid 1990s audience (as Casino Royale did to new audiences of 2006, save for it removed much of Bond's charms).
3) It was a very smart action film. It intelligently handled post-Cold War Russia, Bond's isolation and the cynicism he faced. He is squarely a product of the 20th century and he had to go into the 21st century like it was the dark unknown. That is how the film treated him and he came out on top guns blazing. That makes for a stronger narrative and arc, as he has something to prove in this shadowy, grim world "at the end of history," as neocon ideologues had dubbed the 1990s following the fall of the Berlin Wall. M was a woman, his allies were Russians and his enemy was a left over scar from World War II and Britain's crimes during the covert wars with Russia in the aftermath. And the villain being a fellow double-O agent made it cool.
The moment Bond breaks through a wall in a tank remains the coolest Bond moment of the post-Moore era to date. That is why Goldeneye is so beloved by fans and casual viewers alike. Well that and the awesome 1997 video game, of course.