My great grandfather, Thomas Hamill, served in the Civil War. He enlisted on September 3rd, 1864, right near the end of the war and served less than ten months before being mustered out in June, 1865.  I thought I'd focus on the circumstances surrounding his decision to enlist.
On July 18, 1864, President Abraham Lincoln issued a proclamation calling for 500,000 volunteers in order to supply enough troops, hopefully, to end the Civil War, then in its fourth year.  Newspapers, like the Newark Daily Advertiser, were quick to print the proclamation.  The following image is from the July 19th issue.
A key feature of the call for volunteers was the payment of a bounty. The Federal government offered a bounty of $100 for one year's service as described below in this excerpt from the July 19, 1864, Newark Daily Advertiser. 
Along with the call for volunteers was the threat of a military draft if not enough men signed up. As the proclamation states, "And I hereby proclaim, order and direct that immediately after the fifth day of September, 1864...a draft for troops to serve for one year shall be held in every town, township, ward of a city, precinct, election district, or a county not so sub-divided, to fill the quota assigned to it..." 
The threat of a draft was enough make local jurisdictions scramble to fill their quotas. In Elizabeth, New Jersey, Thomas Hamill's home, the Union County Board of Freeholders voted to pay its own bounty, in addition to the Federal bounty, to volunteers from that county. Other towns in Union County also began offering their own bounties. The article below from August 10, 1864, states that the Third Ward in Elizabeth offered a bounty of $400. 
As I mentioned above, Thomas Hamill enlisted on September 3, 1864, right before the September 5th deadline in the call for volunteers specified by President Lincoln's proclamation. I've already included the front of Thomas Hamill's enlistment document in another post (Decoration Day), but the back of the paper shows Thomas was paid $33.33 as the first installment of his government bounty.  Although I haven't located documentation, I think it's likely that Thomas was also paid a local bounty, either by Union County, New Jersey, or by the City of Elizabeth, or his local ward.
It seems reasonable to assume that Thomas enlisted not only to serve the cause, but to help his family financially. The large and growing Hamill family was poor. The 1860 U.S. census record for the Hamill family shows Thomas, his wife Elizabeth, and six children.  By 1864 a seventh child had been born.  Thomas' occupation is listed as "Day Labor" in the census. Thus, he had no consistent job at that time. The lure of two bounty payments, a Federal one and a local one, which would help feed his family may have been enough to cause Thomas to answer the President's call.
 Thomas Hamill, Pvt., Battery B, 1 N.J. L. Artillery, Volunteer Enlistment, Compiled Military Service Records; Civil War; Records of the Adjutant General's Office, 1780's-1917, National Archives record Group 94; National Archives and Records Administration, Washington, D.C.
 Abraham Lincoln: "Proclamation 116 - Calling for 500,000 Volunteers," July 18, 1864. Online by Gerhard Peters and John T. Woolley, The American Presidency Project. http://www.presidency.ucsb.edu/ws/?pid=69996.
 "Half a Million Men Called for, by the President of the United States of America: A Proclamation; 19 July 1864, p. 2, col. 1; accessed 24 May 2013,Newark Daily Advertiser, Newark, New Jersey, online images (www.genealogybank.com).
 "Half a Million Men Called for..."
[5 Lincoln: "Proclamation 116 - Calling for 500,000 Volunteers."
 "Volunteer Movements in Union County;" 10 August 1864, p. 2, col. 4; accessed 24 May 2013,Newark Daily Advertiser, Newark, New Jersey, online images (www.genealogybank.com).
 Thomas Hamill, Pvt., Battery B, 1 N.J. L. Artillery, Volunteer Enlistment, Compiled Military Service Records, Civil War, RG 94, NA-Washington.
 1860 U.S. census, population schedule, New Jersey, Union , Database online. Census Place: Elizabeth Ward 1; Page: 424; Image: 422; Family History Library Film: 803710. page 86-7 [penned], Thomas Hamul household, dwelling 591, family 716; NARA microfilm publication M653, roll 710; digital image, Ancestry.com (www.ancestry.com); digital image, Ancestry.com (www.ancestry.com).
 1870 U.S. census, population schedule, New Jersey, Union , Database online. Census Place: Elizabeth Ward 2; Page: 378A; Image: 154; Family History Library Film: 552389, Thomas Hamel (Hamill) household, dwelling 355, family 375; NARA microfilm publication M593, roll 890; digital image, Ancestry.com (www.ancestry.com); digital image, Ancestry.com (www.ancestry.com).