Based on your comments on other answers, i would like to provide you a summary on the difference of the CAPM-Alpha and Jensen's-Alpha.
CAPM
The CAPM is an economic model for asset pricing. It states that the equation
$$E[r_i - r_f] = \beta_i E[r_m- r_f]$$
holds for any asset $i$. $r_i$ denotes the return of asset $i$, $r_f$ the risk-free rate of interest, $r_m$ the market-return and $\beta_i$ the beta-factor of asset $i$.
I often hear that the CAPM is just a regression, which is not true in fact (see e.g. this excellent answer here). However, we do run the following regression, when we are empirically testing if the CAPM holds:
$$r_{i,t} − r_{t,f}= \alpha_i + \beta_i (r_{t,m} − r_{t,f}) + \epsilon_{i,t}$$
There are several empirical implications for the CAPM, like excess returns are linear in beta, so coefficients on adding a squared beta-term in the above regression should yield in insignificant coefficients. The main implication however is, that $\alpha_i$ should be indistinguishable from zero for any asset $i$. We test this for multiple assets with an F-test (often called GRS test in finance) or a $\chi^2$-test (see this answer here for further information on the test-statistics).
Empirical evidence shows, that the CAPM is a failure. It just does not work, respectively fails to describe asset returns.
Jensens's alpha
Jensen's alpha is not an economic model, but rather a method to measure portfolio performance. It was first used as a measure in the evaluation of mutual fund managers. How would one measure, if a fund manager has skill or not? Well, let's take a look on the difference of the actual return of a given portfolio and its expected return:
$$\alpha_{i,t} = r_{i,t} - \operatorname{E}[r_{i,t}]$$
That's in fact Jensen's alpha. If it is positive, the fund/portfolio "beats" the expected return and we would assume, that the fund manager has some skill (if the $\alpha_{i,t}$ is significantly different from zero over a longer period of time).
Jensen's alpha and the CAPM together
Well, to calculate Jensen's alpha, one has to figure out the expected return $\operatorname{E}[r_i]$ of a portfolio. We may apply an economic model like the CAPM (or alternatively the market-model, the Fama/French Five Factor Model, etc.) to estimate this expected return. It is up to you to decide, what an appropriate model is. If e.g. a managers stock universe is restricted to S&P500 listed stocks, you could also use the simple S&P500-return as a benchmark for the fund performance (i.e. using $r_t^{S\&P500}$ instead of $\operatorname{E}[r_i]$). Often, the CAPM is used to calculate $\operatorname{E}[r_i]$ when applying Jensen's portfolio measurement index, but now you may see, that this has nothing to do with the $\alpha_i$ when testing the CAPM.
References
Elton/Gruber/Brown/Götzmann (2014), Modern Portfolio Theory and Investment Analysis, ed. 9, John Wiley & Sons.
Fama, E. and MacBeth, J. (1973), Risk, return, and equilibrium: Empirical tests., The Journal of Political Economy, 81(3), pp. 607-636.
Jensen, M.C. (1968), The Performance of Mutual Funds in the Period 1945-1964, Journal of Finance (23), pp. 389-416.