This is probably a latency issue. DAW audio works by filling up buffers with a small amount of PCM-sampled audio, sending that to the computer, which does some processing and mixing, filling up another buffer which is sent to the sound card / interface, which then converts it to analogue that you can hear. So, this response is not speed-of-light like in analogue processing, but does incur this “tiny echo” that you noticed.
Humans, it turns out, are pretty good at both detecting and compensating latency. In your case, what seems to happen is that because you hear your monitor signal delayed, you compensate while performing, so that what you hear back of yourself is on the beat. But the DAW has itself also already latency compensation built in, i.e. on the recording the track will actually be placed at the time you really played/sang – slightly before the beat.
It is possible to change the amount of latency by selecting the buffer size – typically available are 64, 128, 256 ... 2048 samples. The smaller the buffer, the less latency, but it may also make your system unstable. The reason for this problem is that PCs aren't really made for the kind of signal flow that you'd ideally want for audio: processors need to fetch a whole batch of data to run at best efficiency, and the operating system can interrupt the audio thread briefly, which is only ok if the buffer is not yet required (else it's a buffer underflow or overflow, which manifests itself in audio crackle and/or dropout).
So rather than trying to get your DAW latency down, a better solution is zero-latency monitoring: if you hear yourself playing directly, through an analogue or synchronous-DSP signal path, then the problem vanishes. Any decent audio interface offers dry monitoring without latency. If you rely on the computer for guitar amp modelling then it's trickyer. I recommend you find another solution – splitting the signal, using some cheap practice amp for the monitoring and only putting the good effects on the recorded take will probably give better results than if you have great monitor sound but don't get the performance right.