Try NDSolve in this case:

Find the code in the attached notebook.

Attachments:

Thank you so much for your help Shenghui! I had some quick follow up questions: Shenghui, does the need for NDSolve mean that this system cannot be solved analytically for explicit solutions? Additionally, can you explain what the difference between NDSolve and DSolve is and why one works but the other doesn't? Once again thank you so much for your help!

The system has several nonlinearities, which might make it difficult/impossible to find an analytical solution. I think that there was a misinterpretation of the input. In your first post you apparently tried to assign values to the parameters, but you used double equals which indicate an equation rather than an assignment.

a == 15.5, b == 1.3710^-4, c == 1.05 * 10^8, e == 1000

In that case Mathematica does (rightly) not find a solution. Also, you might want to run this:

ClearAll["Global`*"]

sol = ParametricNDSolveValue[{
   m'[t] == -2 a (m[t]^2) + 2 b*d[t] - c*m[t] d[t] + e*o[t], 
   d'[t] == a (m[t]^2) - b*d[t] - c*m[t] d[t] + e*o[t],
   o'[t] == c*m[t] d[t] - e*o[t], m[0] == 1, d[0] == 0, o[0] == 0},
  {m, d, o}, {t, 0, 100}, {a, b, c, e}]

Manipulate[x = sol[a, b, c, e]; 
 LogPlot[{x[[1]][t], x[[2]][t], x[[3]][t]}, {t, 0, 100}], {{a, 
   15.5}, -500, 500}, {{b, 0.0001371}, -1, 1}, {{c, 1.05*10^8}, 10^2, 
  10^10}, {{e, 1000}, -1000, 10000}]

You can modify the values of the parameters and see how your solution changes. Note that for some combinations of parameter values the solution diverges or leads to numerical instabilities.

You wrote

The code itself is as follows:

DSolve[{m'[t] == -2a(m[t]^2) + 2bd[t] - cm[t]d[t] + eo[t], d'[t] == a(m[t]^2) - bd[t] - cm[t]d[t] + eo[t],
o'[t] == cm[t]d[t] - eo[t], m[0] == 1, d[0] == 0, o[0] == 0, a == 15.5, b == 1.3710^-4, c == 1.05 * 10^8,
e == 1000}, {m[t], d[t], o[t]}, t]

I literally scraped your text off the screen, pasted it into Mathematica and both I and Mathematica misunderstood and completely missed the invisible *'s, or perhaps desktop published really really tiny spaces, between bd, cm and eo and so thought, because of the trailing [ t ] that these were three new unknown functions. I apologize for my mistakes.

Not requiring the use of * between variables is sometimes a convenience and all too often a source of errors.

Now with these changes

a = 155/10; b = 13710^-8; c = 105*10^6; e = 1000;
DSolve[{m'[t] == -2 a (m[t]^2) + 2 b*d[t] - c*m[t] d[t] + e*o[t], d'[t] == a (m[t]^2) - b*d[t] - c*m[t] d[t] +
e*o[t], o'[t] == c*m[t] d[t] - e*o[t], m[0] == 1, d[0] == 0, o[0] == 0}, {m[t], d[t], o[t]}, t]

It sits there with the yellow bar letting you know it is thinking and after a while returns the DSolve unchanged, and you are to understand that this means it cannot find an exact analytic solution to your problem. Then as Yang nicely shows, it is possible to get a numerical solution to your problem.