How does a cube sat communicate with ground station with a 1w antenna
Here's how it works but a couple of factoids first: -


According to this source, the most common frequency of transmission is 437 MHz
Antenna gain\$^1\$ for a 1m dish is about \$0.5\cdot\dfrac{(\pi D)^2}{\lambda^2}\$ and at 437 MHz \$\lambda\$ is about 0.69 metres and gain will be about 10 and in dBs this also is 10dB


Link loss from satellite to earth is: -

32.5dB +20log\$_{10}\$(MHz) + 20log20log\$_{10}\$(km)

This is derived from the Friis equations for free-space transmission. At 250km and 437 MHz the link loss is 32.5 dB + 52.8 dB + 47.5 dB = 133 dB. In other words, the power reaching the receive antenna is 133 dB down on what was transmitted. If 1W is transmitted (30 dBm) then expect a receive power level of -103 dBm.

Given that it may be a dish with an antenna gain of 10dB\$^1\$, the power received might be 10 dB higher but how much signal is needed to be received?

What bandwidth/data rate is the received signal? Assume 100kbps (I have no idea what it is but here goes anyway!). Power needed by a receiver to avoid high bit error rates at normal ambient temperatures is: -

-154 dBm + 10log\$_{10}\$(data rate) and for 100kbps this is -104 dBm - beautifully coincidental eh! 

Well, it's a bit worse than that because all sorts of things can interfere so realistically you'd want to see another 10 or 20 dB more received power OR, maybe the data rate is only 10kbps. If it is 10kbps then required received power can be 10 dB lower at -114 dBm and this is about do-able even without a parabolic dish at the satellite. It also assumes a low gain dipole as the receiver but there is no reason why you shouldn't use a dish if the satellite is geostationary. You'll get a couple of advantages if you do; first the received gain will be higher and 2nd, the dish is collecting power from space and the background level is much lower because space is cold. This would make the equation immediately above this paragraph much more lenient towards the receiver.

EDITED to put the negative sign in front of a couple of dBm numbers!!!

\$^1\$ I'm just seeing if it needs a dish so I thought I'd preempt the calculation up-front.