The formal announcement came around 10 a.m. C-SPAN will have video of the nomination available, here, shortly.
In his introduction, the president stressed her background (Compelling Personal Narrative Alert), her ability to understand ordinary people (Empathy Alert), and the fact that he "relishes" the idea of having three women on the Court for the first time ever.
Phil Klein has a good write-up of Obama's and Kagan's short remarks.
The Federalist Society has set up the site, SCOTUS Report, for frequent updates on the nomination process and fight.
Conservative legal experts react to the nomination, here.
The always thorough SCOTUSblog is live-blogging the day's news.
C-SPAN has an archive page of every single one of Kagan's C-SPAN appearances as Solicitor General.
The Daily Caller notes a Kagan quote about nomination process that may come back to haunt her: "Elena Kagan in 1995 argued that Supreme Court nominees should have to answer direct questions from senators about specific issues and pending cases, criticizing a confirmation process that avoids direction questioning as a 'vapid and hollow charade.'"
Glenn Greenwald applies the ouch, from the left: "It's anything but surprising that President Obama has chosen Elena Kagan to replace John Paul Stevens on the Supreme Court. Nothing is a better fit for this White House than a blank slate, institution-loyal, seemingly principle-free careerist who spent the last 15 months as the Obama administration's lawyer vigorously defending every one of his assertions of extremely broad executive authority. The Obama administration is filled to the brim with exactly such individuals -- as is reflected by its actions and policies -- and this is just one more to add to the pile."
More doubts from the left, at Mother Jones: "Kagan told the Georgetown audience that the [Kennedy] had 'a bit of a bad habit,' namely that he asks advocates about cases that are not mentioned anywhere in the briefs for the case. Kennedy did just that in Citizens United when he asked Kagan whether something she had just said was 'inconsistent with the whole line of cases that began with Thornhill v. Alabama and Coates v. Cincinnati.' Perhaps many advocates know those cases, Kagan said, but 'I at any rate did not.' She added, 'There was a look of panic on my face.'"