Looks like a weeping ornamental cherry. I would allow it to weep - not cut it straight as shown in the picture. If you need the branches to be higher, i.e. in order to mow under it, perhaps pruning it up higher so that the new branches don't grow so far down. Wherever you cut, normally one or two new branches will grow. Prune after it goes dormant (leaves fall).
That looks like a Weeping Cherry Tree