Petersburg is incredibly complicated, even trying to name the various engagements proves tricky. The easiest way to keep it clear is to break it down into a series of campaigns and offensives.
The first fighting at Petersburg predates the "Petersburg Campaign" itself. During Ben Butler's Bermuda Hundred Campaign, an expeditionary force from the Army of the James crossed the Appomattox River to test Petersburg's defenses before being pushed back in what became known as the Battle of Old Men and Young Boys, June 9, 1864. This is sometimes referred to as the First Battle of Petersburg.
The Army of the Potomac arrived from Cold Harbor the following week and launched attacks against the eastern defenses for four days, June 15-18. This is the first of Ulysses S. Grant's offensives against the Cockade City, but it is typically called the Second Battle of Petersburg.
Failing to take Petersburg through direct assault, Grant sought to cut the supply lines into the city and launched his Second Offensive, June 21-24, also known as Jerusalem Plank Road and First Weldon Railroad. At the same time he sent a portion of his cavalry into southern Virginia to raid the South Side Railroad and Richmond & Danville Railroad. The Wilson-Kautz Raid, June 22-July 1, was turned back at the Battle of Staunton River Bridge, June 25.
Next came the infamous Crater, July 30, but the explosion and resulting failed charge was part of a larger strategy by Grant to lure Lee into sending his men back north of the James River to protect Richmond. The Third Offensive, July 27-30 also included the fighting at First Deep Bottom. It should be noted that after the Crater, Grant won't order his men to attack fixed Confederate fortifications again until April 2, 1865.
His strategy, instead, reverts to his goal at the end of June, and the Fourth Offensive, August 14-25, involves another attempt to draw Confederates to Richmond and away from Petersburg (Second Deep Bottom), cutting another supply line (Globe Tavern or Second Weldon Railroad), and extending the damage to that railroad (Reams Station).
Every time the Union army gobbles up terrain around Petersburg, they extend their fortifications. The Confederates cause a scare the next month with the Beefsteak Raid (September 14-17), but it has little effect besides a temporary morale shift.
Grant goes back to his useful strategy again with the Fifth Offensive, September 29-October 13, capturing Confederate positions southeast of Richmond at New Market Heights and Fort Harrison, but Confederate defense at Chaffin's Farm protects Richmond itself. Meanwhile, south of Petersburg, the Army of the Potomac overruns a newly built Confederate line at Peebles Farm, but can't reach their actual targets (Boydton Plank Road and the South Side Railroad).
The Union try again with a Sixth Offensive (October 27-28) that reaches the Boydton Plank Road at Burgess Mill before forced to withdraw back to the ground gained (and since fortified) around Peebles Farm. There is also fighting east of Richmond during the month, mainly small expeditions and cavalry raids, at Darbytown Road and Second Fair Oaks.
Though Union forces cut the Petersburg & Weldon Railroad in August, the Confederates continue to use it up until a point about twenty miles south of the city (and past Union control). Grant sends a large raid south along the Petersburg & Weldon Railroad supply line, hoping to destroy it all the way to North Carolina. This Hicksford Raid (December 7-12) is also known as the Stony Creek Raid or Applejack Raid.
Both armies then settle down for a few months, but a temporary break in the winter weather encourages Grant to try to cut the Boydton Plank Road again, resulting in the inconclusive battle of Hatcher's Run, or Dabney's Mill, February 5-7, 1865.
Lee's prospects continue to whittle away, as Confederate forces elsewhere rack up defeats while the Army of Northern Virginia is stuck protecting Petersburg. Desperate to force Grant to contract his ever-tightening noose around the city, Lee attacks the Union fortifications (something he hadn't tried since June) but is defeated at Fort Stedman, March 25, 1865. Southwest of the city, the Union Second and Sixth Corps capture the CS picket line that afternoon at Watkins Farm and Jones Farm.
Grant's final offensive against Petersburg is sometimes lumped in with the Appomattox Campaign. Union columns continue the westward push around the Confederate lines and fight at Lewis Farm (March 29), White Oak Road (March 31), Dinwiddie Court House (March 31), and Five Forks (April 1). On April 2, Union forces attack all along the Petersburg front. This assault is sometimes referred to as the Third Battle of Petersburg, which is ridiculous considering the amount of fighting done since the "Second Battle" in June. The Ninth Corps attacks at Fort Mahone, the successful Sixth Corps charge didn't really have a proper name at first but is now known as the Breakthrough, the Second Corps fights at White Oak Road and Sutherland Station, portions of the Twenty-fourth Corps fight at Hatcher's Run and later Fort Gregg. By early afternoon, Lee has instructed his army to evacuate both Petersburg and Richmond and its on to Appomattox.
So, yeah, a lot of different names for the fighting around Petersburg.