Morgan Stanley is buying E*Trade Financial Corp. in a $13 billion deal that will reshape the storied investment bank and firmly stake its future on managing money for regular people.

The all-stock takeover, announced Thursday, will combine a Wall Street firm in the late innings of a decadelong turnaround with a discount broker built on the backs of dot-com day traders. It is the biggest takeover by a giant U.S. bank since the 2008 crisis.

...