I'll limit the answers to character reasons, and mainly to the main characters. November, February and the penultimate episode of the season in May are the popular exit episodes. In order of departure:
Dr. Susan Lewis (Sherry Stringfield) first left in episode "Union Station" (3.8, November 21, 1996). She left to be with her sister and niece Suzie in Phoenix, Arizona. She returned in episode "Never Say Never" (8.4, October 18, 2001)- Dr. Dave Malucci's (Erik Palladino) last episode. She left the second time in season 12's premiere episode "Canon City" (September 22, 2005) after accepting a tenure track position in Iowa, after being denied at County (offscreen).
Chief of emergency medicine, Dr. David Morgenstern (William H. Macy), resigned in episode "Shades of Gray" (4.19, April 23, 1998). Returning to work 6 months after a heart attack (in season's 4 opening episode), Morgenstern finds it difficult coming back. During a surgery Dr. Peter Benton pushes him aside, but he's unable to save the patient. Peter is suspended (for the shoving), but David reinstates him before resigning.
Dr. Anna Del Amico (Maria Bello) was last seen in season's 4 finale episode "A Hole in the Heart" (May 14, 1998). In the first episode of season 5, Carter explains that she moved to a pediatric ER in Philadelphia, where her family lives.
Dr. Doug Ross (George Clooney) was caught stealing pain medication from a double blind study and is put on probation by Mark and Kerry hoping to cover up his actions, which if got out could devastate the hospital's funding. Scandal erupts when Doug shows Joi Abbot (Valerie Mahaffey) her how to bypass the lockouts on a PCA machine and she then gives her terminally ill son an overdose on Dilaudid. Facing possible criminal charges in the boy's death the hospital suspends Ross from treating patients. With his career now in shambles Doug formally resigns and relocates to Seattle. He asks Carol to leave with him, but she declines. "The Storm (part 2)" (5.15, February 18, 1999) One year later Carol and Doug reunite when she leaves Chicago to be with him. 10 years later, Doug now happily married to Carol is seen working in Seattle as a Peds Attending.
Jeanie Boulet (Gloria Reuben) has two brief scenes in episode "The Peace of Wild Things" (6.6, November 11, 1999). She left the ER to spend more time with her husband, Officer Reggie (Cress Williams), and their HIV positive foster child Carlos.
Lucy Knight (Kellie Martin) was stabbed to death by a mental patient (David Krumholtz) in episode "All in the Family" (6.14, February, 2000).
Nurse Carol Hathaway (Julianna Margulies) was last seen in episode "Such Sweet Sorrow" (6.21, May 11, 2000). She leaves the ER and Chicago after treating a dying patient and consoling her family. Carol decides that she must give herself one last shot at happiness and flies to Seattle to reunite with Doug. It's later revealed by Halleh that Carol sent for the twins soon afterwards. 10 years later we see Carol happily married to Doug and working as a Transplant coordinator in Seattle.
Dr. Peter Benton (Eriq La Salle) briefly appeared in 2 additional episodes after his original final episode "I'll Be Home for Christmas" (8.10, December 13, 2001) in which he got custody of his son Reese and moved to the suburbs with Dr. Cleo Finch (Michael Michele) who also appeared in an episode after this supposedly last episode.
Dr. Mark Greene (Anthony Edwards) died from a brain tumor in Hawaii "On the Beach" (8.21, May 9, 2002).
Dr. Robert Romano (Paul McCrane) was killed when a helicoptor crashed and landed on him, causing an explosion, in episode "Freefall" (10.8, November 20, 2003). Dr. Kerry Weaver was sorry to see him go, but no one else in the show was.
EDIT: It wasn't Dr. Kerry Weaver, it was Dr. Elizabeth Corday Greene.
Dr. Michael Gallant (Sharif Atkins) was sent to Iraq in episode "Where There's Smoke" (10.18, April 8, 2004). He returned for a 72 hours leave in 2 episodes on season 11. He died in an explosion in Iraq in the episode "The Gallant Hero and The Tragic Victor" (May 11, 2006). Edit: Gallant was indeed in two episodes in season 11, but he was only home for one of those episodes "Back in The World". The other episode, he was still in Iraq. Also, he returned for four episodes in season 12 (including "TGHATTV") in which he marries Neela and then decides to go back to Iraq.
Dr. Elizabeth Corday (Alex Kingston) returned to England in episode "Fear" (11.4, October 21, 2004) after rejecting Kerry's offer to be a clinical instructor. She did so when Corday was on the brink of losing her medical license after performing an illegal organ donation procedure.
Medical Student, Deb Chen (Ming-Na Wen) left medical school in episode "House of Cards" (1.21, April 6, 1995) after almost killing a patient doing a procedure she shouldn't have done unsupervised. She returned in episode "Family Matters" (6.10, January 6, 2000), now going by Jing-Mei instead of 'Deb'. Her final episode was "Twas the Night" (11.9, December 9, 2004) in which she euthanizes her father.
After getting tenure, Dr. John Truman Carter III (Noah Wyle) leaves in season 11's finale "The Show Must Go On" (May 19, 2005) for Africa, to be with Kem (Thandie Newton). He appears during season 12 in the episodes "Darfur", "No Place to Hide" and "There are no Angels Here". He also returns near the end of season 15 in the episode "The Beginning of The End" and stays on through the end of the series.
Dr. Kerry Weaver (Laura Innes) leaves the ER in "A House Divided" (13.13, January 11, 2007) after accepting a position in medical news broadcasting in Miami, Florida.
Dr. Ray Barnett (Shane West) leaves in "The Honeymoon is Over" (13.23, May 17, 2007) after being hit by a truck while drunk. He got his legs amputated, and now works in physical rehabilitation.
After being hurt in an ambulance explosion, Dr. Greg Pratt (Mekhi Phifer) dies in the ER in the season 15 premiere "Life After Death" (15.1, September 25, 2008).
Dr. Luka Kovac (Goran Visnjic) officially quit his job in season 14, and made guest appearances throughout the season being Abby's husband. Dr. Abby Lockhart (Maura Tierney) and Luka leave Chicago together in "Book of Abby" (15.3, October 16, 2008) to start a fresh life in Boston with their son Joe.
Dr. Abby Lockhart leaves in "Book of Abby" with Luka and son Joe to start over in Boston. However, she makes a brief appearance in "Shifting Equilibrium", giving Neela advice over the phone when Neela wavers over leaving County.
Dr. Neela Rosgotra leaves County after realizing that she's still in love with Ray. She moves to Baton Rouge to take a teaching position in a University Hospital near where Ray works in rehabilitation. She makes a brief appearance in the next episode "I Feel Good" via a webcast with County.