OFDMA is an extension of OFDM from a modulation method to a multiple-access method, and it is used for the same reasons that OFDM is used, i.e., mainly provides high data rate and at the same time eliminates ISI without the need of complex channel equalizers and channel estimation - something very important in wireless communications with fading where the channel variations are often very rapid.
It would be great if we could use OFDMA not only in the DL but also in the UL. However, OFDMA suffers from high peak-to-average power ratio (PAPR), due to the shape of the transmitted signal. This requires a large back-off of the power amplifier at the transmitter in order to ensure that it will produce linear output. But, as you know, UL is the communication channel from the user equipment towards the base station (in case of LTE, the eNodeB). Thus, the use of OFDMA in the UL would increase the transmitting power requirements and would reduce the battery life of the mobile phone (or PDA or whatever), and surely we don't want that.
So, what is the alternative for the UL? SC-FDMA, which, despite of its name, it is not single-carrier at all! In SC-FDMA each sub-carrier is modulated with a linear combination of modulation data symbols (e.g., QAM symbols). In other words, with SC-FDMA data is transmitted over the air interface in many sub-carriers but adds the additional pre-processing step to build a linear combination from the modulation symbols. This linear combination is provided via DFT, and for that reason this technique is often called DFT pre-coding. The advantage of SC-FDMA over OFDMA is that different sub-carriers have different amplitudes, so the PAPR in the time-domain can be as low as that of the QAM data symbols.
To sum up:
In OFDMA each sub-carrier only carries information related to ONE specific symbol.
In SC-FDMA each sub-carrier contains information of ALL transmitted symbols. This is done to create a signal with low PAPR, as required in the UL in order to keep the battery life of UE high.
The following image I think illustrates the above much better from any words:
You can take also a look on the OFDMA transmitter-receiver and the SC-FDMA transmitter-receiver (you will find plenty of info and images on the internets about that) to see their differences (main difference, as I said, is that in SC-FDMA there is first a FFT, then a sub-carrier mapping and only then an IFFT to create the time-domain signal, whereas in OFDMA there is no DFT-precoding).
I hope I helped you. It is a complex subject though and no answer in a forum can be 100% accurate. So, remember the main point: We cannot have OFDMA in UL due to the high PAPR - thus, we use SC-FDMA which is like OFDMA with DFT precoding! Simplified, but pretty-much true ...