I think individuals are forced to retire because of privatizations in public sector, or they should retire or to retire in order to lose their health because of the high level of work-related work accidents and occupational illnesses. The most important factor in making a decision to work after a person retires lies in meeting the income needs of the family. After retirement, the most important factor in the decision of non-employment of the individuals is the life of health problems.
The financial and health situation certainly plays a role. Retirees with low pension income and wealth are more likely to search for a job after retirement in order to support themselves. Workers in poor health are less likely to want to work post retirement (although high health cost and employer health insurance might tempt them). However, not everybody who wants to continue working finds a job (https://doi.org/10.1093/geront/gnu182)
There are many, but I think it depends on the level of analysis or whether you are examining across levels. For example there are policy factors and broader economic factors involved within a country (rate of unemployment, public pension coverage). There are also life course factors to consider (e.g., whether the retirement was voluntary or involuntary, the individual’s age, marital status, and health). Education and socioeconomic status play a role as well. Organizational hiring policies and the demands or nature of the work are involved too. Working in retirement also involves the individual’s motivations for working and opportunity cost with other activities.
There are many factors. Important ones are voluntary/involuntary retirement, individual age, family status, health condition, household income; also to be considered professional profile, socio-economic status, social capital, labour market features/opportunities
Apart from more socio-economic reasons, there are also some psychological factors to consider, especially the individual attitudes towards work. If you are interested in these factors, here are some good papers to start with:
Fasbender, U., Wang, M., Voltmer, J.-B., & Deller, J. (2016). The meaning of work for post-retirement employment. Work, Aging and Retirement, 2, 12-23. doi:10.1093/workar/wav015
Kim, J.E., & Feldman, D.C. (2000). Working in retirement: The antecedents of bridge employment and its consequences for quality of life in retirement. The Academy of Management Journal, 43, 1195-1210. doi: 10.2307/1556345
Wang, M., Zhan, Y., Liu, S., & Shultz, K. (2008). Antecedents of bridge employment: A longitudinal investigation. Journal of Applied Psychology, 93, 818–830. doi:10.1037/0021-9010.93.4.818
Wöhrmann, A. M., Fasbender, U., & Deller, J. (2016). Using work values to predict post- retirement work intentions. Career Development Quarterly, 64, 98-113. doi: 10.1002/cdq.12044