Thank you so much for your help Shenghui and Bill! Other solutions welcome as well!

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 yellow bracket means it is busy thinking, and for some problems it has to think a long time.

Issues: You have 6 unknown functions, m, bd, cm, d, eo and o. You only have 3 equations and 3 initial conditions. If you remember algebra from long ago, if I gave you a simple problem with 6 unknowns, but only 3 equations you can't find a solution for all 6 of those unknowns.

Here is a slight rewrite of your problem to make it a little more acceptable to Mathematica

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

Sometimes DSolve can find answers if it has exact fractions instead of numbers with decimals. But that didn't make enough of a difference and still doesn't solve the problem that there are too many unknown functions to find.

Is there more known information about the problem?

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.