Obviously Liverpool are our main source of rivalry. Before the Premiership started, Liverpool were by far the best team in the land, enjoying the same kind of dominance as United do today. What made things worse was when Everton had a good period during the 80's, only for Liverpool fans to get us kicked out of Europe. This hurt us badly and our team made their separate ways, but now Liverpool are much more within reach, so the rivalry is probably as big now as it ever was.
Because Liverpool and Manchester are the 2 biggest Cities in the North West of England, there are cross city rivalries too. Liverpool and United in particular, being the top 2 clubs in the country historically, but when ever we play Manchester United or Manchester City, it's generally a big game to win.
Liverpool, United, and City are now our biggest games of the season in terms of local pride.
In terms of challenging for the spoils in the Premier League, there becomes a different kind of rivalry, as England tends to be split up into North West (as above), North East (Newcastle, Sunderland, Hull etc), The Midlands, containing such teams as West Bromwich Albion, Wolverhampton Wanderers, Birmingham City, Aston Villa etc, and the South East (London), such as your Spurs, Arsenal, Chelsea, West Ham, Fulham etc.
We tend not to worry too much about the North East sides, or the East / West Midlands sides in terms of rivalry, although Villa would stand out historically as a huge team, but the teams in the South, more so the London area, than anywhere else become a want to win game again, as not only is there a big North South divide in this country, but the main football powers have always been from the cities of Liverpool, Manchester, and London.
Of the London sides, we are probably more determined to enjoy victory over Arsenal, more than the others, as they are the best historically of the Southern sides.
Before the bankrolling of teams, I'd say the biggest teams in the country were Liverpool, United, and then Arsenal. Everton would probably come next in terms of league titles, and although Villa and Sunderland have won many titles between them, they tend to be less successful since the war.
With the bankrolling of teams, Chelsea and Manchester City, and probably even Spurs have pretty much paid for success with big wallets, so beating them is also satisfying because of that.
My own personal 'biggest' matches are Liverpool and United.