How many mg caffeine is 1mM caffeine?

I am not sure what is confusing the original poster. The comments were clear and the answer by Jan is clear. Let's try to walk you through the problem again.

The question asks about how many milligrams are there in $\pu{1 mM}$ caffeine solution. Did you attempt a dimensional analysis?

For mass to mole conversions (or the other way you round) you need the molecular weight. The molecular weight has units of $\pu{g/mol}$. Caffeine's molecular weight is $\pu{194.19 g/mol}$.

$\pu{1 mM}$ (read as one millimolar) of any substance $X$ means that $\pu{1E-3 mol}$ of $X$ is present in $\pu{1 L}$ of solution.

Amount of substance (unit: $\pu{mol}$) to mass (unit: $\pu{kg}$) conversion requires the knowledge of the molecular weight.

$$\pu{1E-3 mol} \times \pu{194.19 g/mol} = \pu{0.19419 g}$$ (Note that amount of substance units cancel.)

Convert grams to milligrams by recalling the definition that $\pu{1 mg} = \pu{0.001 g}$. This means that $\pu{194.19 mg}$ of caffeine are present in $\pu{1 L}$ solution. Round off to desired significant figures. Note never round off during calculations. (Hint: $\pu{1 mM}$ has only one significant figure).

Also note: mMol is not an acceptable notation, it better be $\pu{mmol}$ to be read as millimole. This is not the same as $\pu{mM}$ which is read as millimolar. The former is an amount unit, the latter is a concentration unit.

To my understanding, M. Farooq correctly address the OP's problem. As noted in his answer:

$\pu{mMol}$ is not an acceptable notation, it better be $\pu{mmol}$ to be read as millimole. This is not the same as $\pu{mM}$, which is read as millimolar. The former is an amount unit, the latter is a concentration unit.

That's it. OP needs to understand that units $\pu{mmol}$ and $\pu{mg}$ are amount units, while $\pu{mM}$ and $\pu{M}$ are concentration units (by convention, $\pu{M}$ is equal to $\pu{mol\:L-1}$).

Let's look at OP;s problem now:

Wikipedia says that the molar mass of caffeine is $\pu{194 g/mol}$. (1) Does that mean that $\pu{1 mol}$ caffeine equals $\pu{194 g}$ of caffeine? (2) And that $\pu{1mM}$ caffeine, in turn, equals $\pu{194 mg}$ caffeine?

Answer to (1) is yes: OP must read and understand two already given answers elsewhere.

Answer to (2) is no: As M. Farooq correctly pointed out (vide supra), $\pu{mM}$ is a concentration unit (of a solution) and $\pu{mg}$ is an amount unit of a substance (e.g., it could be about substance in the solution such as caffeine in given coffee solution). They do not equal to each other ever without knowing another dimension called volume. Only way to make relationship between these two unit ($\pu{mg}$ and $\pu{mM}$, etc.) is to know how much volume we are considering.

For example, OP can ask the question correctly as: How many $\pu{mg}$ of caffeine in $\pu{1 L}$ of coffee solution, if it is $\pu{1mM}$ in caffeine?

Thus, the amount of caffeine $= \pu{1 L} \times \frac{\pu{1 mmol}}{\pu{1 L}} \times \frac{\pu{198 mg}}{\pu{1 mmol}} = \pu{198 mg}$

If the question is: How many $\pu{mg}$ of caffeine in $\pu{250 mL}$ of coffee solution, if it is $\pu{0.75 mM}$ in caffeine?

Then, the amount of caffeine $= \pu{250 mL} \times \frac{\pu{1 L}}{\pu{1000 mL}} \times \frac{\pu{0.75 mmol}}{\pu{1 L}} \times \frac{\pu{198 mg}}{\pu{1 mmol}} = \pu{37.1 mg}$

Thus, bottom line is if you know the concentration of caffeine in any drink, you must know how much you are consuming (by volume) in order to calculate your caffeine intake (in $\pu{mg}$, etc.).

T'd like to finish this answer with some interesting data. According to Wikipedia:

Depending on the type of coffee and method of preparation, the caffeine content of a single serving can vary greatly. The caffeine content of a cup of coffee varies depending mainly on the brewing method, and also on the coffee variety. According to the USDA National Nutrient Database, an 8-ounce ($\pu{237 mL}$) cup of "coffee brewed from grounds" contains $\pu{95 mg}$ caffeine, whereas an espresso ($\pu{25 mL}$) contains $\pu{53 mg}$.

Thus, the concentration of caffeine in brewed coffee $= \pu{95 mg} \times \frac{\pu{1 mmol}}{\pu{198 mg}} \times \frac{1}{\pu{237 mL}}\times \frac{\pu{1000 mL}}{\pu{1 L}} = \pu{2.02 mmol\:L-1}$

On the other hand, the concentration of caffeine in espresso $= \pu{53 mg} \times \frac{\pu{1 mmol}}{\pu{198 mg}} \times \frac{1}{\pu{25 mL}}\times \frac{\pu{1000 mL}}{\pu{1 L}} = \pu{10.71 mmol\:L-1}$

That means espresso is almost 5 times stronger than brewed coffee (by volume)!