In lecture 22 of Lurie's notes on chromatic homotopy theory there is the following cryptic definition.
For each integer $k$, let $M(k)$ denote the cofiber of the map $Σ^{2k} \mathrm{MU}(p) \to \mathrm{MU}(p)$ given by multiplication by $t_k$.
Here, $\mathrm{MU}(p)$ is mod-$p$ complex bordism, which I presume means the localization of $\mathrm{MU}$ at $\mathrm{H}\mathbb{F}_p$, and $t_k$ is a generator of $\pi_{2k}(\mathrm{MU}(p))$.
Question. What does this mean?
Some thoughts: I can interpret $\Sigma^{2k} \mathrm{MU}(p)$ as $\Sigma^{\infty}S^{2k} \wedge \mathrm{MU}(p)$, so we can consider a map $\Sigma^{\infty} S^{2k} \wedge \mathrm{MU}(p) \to \mathrm{MU}(p) \wedge \mathrm{MU}(p)$ defined by, say, $t_k \wedge \operatorname{Id}$, followed by some kind of multiplication map on $\operatorname{MU}(p)$. I believe $\operatorname{MU}$ is a ring spectrum, which I think implies that $\mathrm{MU}$ admits a multiplication map of some sort. Maybe that's what's being implied here.
The spectrum $MU$ is a ring spectrum. This means that it comes a equiped with a 'product' map $\mu:MU\wedge MU\rightarrow MU$ as well as a unit $\eta:S\rightarrow MU$, which together make all the obvious diagrams (homotopy) commute. Then if $x\in\pi_rMU$, multiplication by $x$ is the composite
$$\hat x:S^r\wedge MU\xrightarrow{x\wedge 1}MU\wedge MU\xrightarrow{\mu}MU.$$
It is standard to drop the hat and denote the induced map by the same symbol as the element $x\in\pi_rMU$. Thus $t_k\in\pi_{2k}MU$ is a generator and the map you are interested in is what I have called $\hat{t}^k$ in the previous notation.
In response to your second question, why are you learning homotopy theory? If it is beacuse you want to become a homotopy theorist, then I'm afraid you will need to put up with the jargon, since this is the language in which we have come to communicate. One point of advice I would offer is to not start with Lurie's notes if this is your first introduction to the subject. They are beautifully written, but as you have found with your question, are not for the beginner. Moreover they are the transcripts of a set of lectures, rather than a dedicated text, and as it always is, there is more knowledge imparted during the actual lecture than is collected on paper afterwards. If you are leaning homotopy theory as a complement to some other subject, then you may do well to try out a text written by someone with a closer background to yours.